What Is the Radius of Continuity in the Function Space F(R, R)?

1. Introduction

1.1. Real-Valued Functions and Continuity

We work throughout in the function space $F(\mathbb{R},\mathbb{R})$ of all real-valued functions defined on the real line. Within this space, continuity remains one of the central regularity properties. In standard real analysis, a function $f:\mathbb{R}\to \mathbb{R}$ is said to be continuous at a point $x_0\in \mathbb{R}$ if for every $\epsilon >0$ there exists a $\delta >0$ such that for all $x\in \mathbb{R}$

$$\begin{array}{c}\hfill |x-x_0|<\delta ⟹|f\left(x\right)-f\left(x_0\right)|<\epsilon .\end{array}$$

(1)

This $\epsilon$–$\delta$ definition, and its various equivalent formulations in terms of sequences or inverse images of open sets, provides the standard language for studying local behaviour of real-valued functions [1,2].

A stronger notion, uniform continuity on a set $E\subseteq \mathbb{R}$, requires that for each $\epsilon >0$ there is a single $\delta >0$ that statement (1) works simultaneously for all points $x_0\in E$. Uniform continuity is closely related to the classical modulus of continuity of a function, which quantifies the size of oscillations at a given spatial scale and plays an important role in analysis on metric spaces and in regularity theory [3].

In the present paper we remain in this classical setting—real-valued functions on $\mathbb{R}$ equipped with the Euclidean metric $d_E$—but shift the emphasis from the qualitative statement “f is (uniformly) continuous” to a quantitative description of how far one can move in the domain while preserving a prescribed $\epsilon$-accuracy in the range. Our goal is to refine the usual continuity framework by associating to each function a family of radii that encode the local and global “size” of its continuity.

1.2. Motivation

The $\epsilon$–$\delta$ definition of continuity is inherently quantitative: given $\epsilon >0$ and a point $x_0$, any admissible $\delta >0$ determines a neighbourhood on which f fluctuates by at most $\epsilon$. However, in most textbooks and classroom proofs one only exhibits some convenient $\delta =\delta (\epsilon ,x_0)$; the collection of all admissible radii is rarely examined systematically [1,2]. This leaves several natural quantitative questions unanswered. For instance: for a fixed accuracy level $\epsilon$, what is the largest radius around $x_0$ on which the $\epsilon$-tolerance still holds? How does this maximal radius depend on $x_0$ and on the structure of f?

In earlier work, the author proposed a global classification of the function space $F(\mathbb{R},\mathbb{R})$ into blocks according to the cardinalities of the sets of continuity and differentiability points [4]. That framework was at macroscale and qualitative: it distinguished functions by the size and location of their continuity sets but did not deal with it in microscale nor quantify the strength of continuity at individual points or uniformly on $\mathbb{R}$. In the follow-up work, the author surveys the $ϵ-\delta$ definition for the classes of continuous functions at microscale level [5]. The present paper aims to complement that perspective by introducing and studying the radius of continuity at a point and its uniform counterpart.

More precisely, for a given function f, point $x_0$, and accuracy level $\epsilon >0$, we consider the set of all radii $\delta >0$ for which the $\epsilon$–$\delta$ condition holds, and define the pointwise radius of continuity${\delta }_f(\epsilon ,x_0)$ as the supremum of this set. Analogously, we define the uniform radius of continuity${\delta }_f^U\left(\epsilon \right)$ by taking the largest radius that works simultaneously for all $x_0\in \mathbb{R}$. These radii can be expressed in terms of pointwise and uniform oscillation functions, thereby connecting our viewpoint to classical moduli of continuity [3,6,7,8].

This radius-based approach has several advantages. It packages the infinitary quantifiers in the usual continuity definition into concrete numerical invariants; it allows for explicit closed-form computations in important families of functions (for example, quadratic polynomials or normal density functions); and it offers a way to compare the “stability” of different functions under perturbations of the input. At a more conceptual level, radii of continuity provide a bridge between textbook $\epsilon$–$\delta$ arguments and quantitative notions that are commonly used in modern analysis and numerical applications, where explicit control over neighbourhood sizes is often required.

1.3. Organization of the Paper

The remainder of the paper is organized as follows. In Section 2 we introduce the pointwise and uniform radii of continuity, develop equivalent descriptions in terms of local and uniform oscillation, and establish basic structural properties. In particular, we characterize non-constant and constant functions in $F(\mathbb{R},\mathbb{R})$ through the finiteness or infiniteness of their associated radii. Section 3 is devoted to explicit computations: we derive a closed-form formula for the pointwise radius of continuity of a quadratic polynomial and analyze the uniform radius of continuity for the normal probability density function, illustrating how the abstract theory translates into concrete expressions. Finally, Section 4 concludes the paper with a discussion on the main findings and outlines several directions for future work.

2. The Radius of Continuity: Definitions & General Properties

2.1. Radius of Pointwise Continuity

In this section, we introduce, for each function f, point $x_0\in \mathbb{R}$, and accuracy level $\epsilon >0$, the set ${\Delta }_f(\epsilon ,x_0)$ of all radii $\delta >0$ for which the usual $\epsilon$–$\delta$ condition holds at $x_0$, and we define the pointwise radius of continuity ${\delta }_f(\epsilon ,x_0)$ as the supremum of this set (Figure 1). This quantity admits an equivalent description in terms of the local oscillation of f[6,7] on the symmetric interval $(x_0-\delta ,x_0+\delta )$, or, after a shift of variables, in terms of an h-dependent oscillation around the origin. We then collect a number of elementary but useful properties of ${\delta }_f$, including an evenness property in the spatial variable, natural scaling relations, and inequalities that arise when f is globally Lipschitz. Further bounds describe how radii behave under algebraic operations: the radius for a sum or product of functions can be estimated in terms of the radii of the individual factors, and similar control holds for compositions under mild regularity conditions. These results show that the radius of continuity is compatible with the standard algebraic operations in $F(\mathbb{R},\mathbb{R})$, and can therefore be propagated through many common constructions. Finally, we prove that the finiteness pattern of ${\delta }_f(\epsilon ,x_0)$ characterizes whether f is constant or not: a non-constant function necessarily has finite radius at some continuity point, whereas a function with infinite radius everywhere must be constant.

Definition 1

(Radius of Pointwise Continuity). Let $f:\mathbb{R}\to \mathbb{R}$ be a real-valued function with a nonempty set of continuity points $C\left(f\right)\ne ⌀$. For $x_0\in \mathbb{R}$ and $\epsilon >0$ define the pointwise admissible continuity admissible radius as:

$$\begin{array}{ccc}\hfill {\Delta }_f(\epsilon ,x_0)& :=& \left\{\delta >0:\forall x\left(\right|x-x_0|<\delta \Rightarrow |f\left(x\right)-f\left(x_0\right)|<\epsilon )\right\},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\delta }_f(\epsilon ,x_0)& :=& sup{\Delta }_f(\epsilon ,x_0).\hfill \end{array}$$

(3)

Remark 1

(Shifted-h formulation of ${\Delta }_f$ and ${\delta }_f$). Equivalently, in terms of the local oscillation

$$\begin{array}{c}\hfill {\omega }_f(x_0;\delta ):=\underset{\left|h\right|<\delta }{sup}|f(x_0+h)-f\left(x_0\right)|,\end{array}$$

(4)

we have:

$$\begin{array}{ccc}\hfill {\Delta }_f(\epsilon ,x_0)& :=& \{\delta >0:{\omega }_f(x_0;\delta )<\epsilon \},\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\delta }_f(\epsilon ,x_0)& :=& sup\{\delta >0:{\omega }_f(x_0;\delta )<\epsilon \}.\hfill \end{array}$$

(6)

Proposition 1

(Properties of Radius of Pointwise Continuity). Let $f:\mathbb{R}\to \mathbb{R}$. Then, for $\epsilon >0$ and $x_0\in \mathbb{R}$ the radious of pointwise continuity ${\delta }_f(\epsilon ,x_0)$ has the following properties:

(A) Single–function properties.

(i) 

Evenness. If f is even, then ${\delta }_f(\epsilon ,x_0)={\delta }_f(\epsilon ,-x_0)$.

(ii) 

Scaling. If $c\ne 0$, then ${\delta }_{cf}(\epsilon ,x_0)={\delta }_f(\epsilon /|c|,x_0)$; if $c=0$, then ${\delta }_{0·f}(\epsilon ,x_0)=+\infty$.

(iii) 

Large–$\epsilon$ for bounded f. If $\left|f\right|\le M$ everywhere and $\epsilon \ge 2M$, then ${\delta }_f(\epsilon ,x_0)=+\infty$ for all $x_0$ (the constant $2M$ is sharp in general).

(iv) 

Monotonicity in $\epsilon$.If $0<{\epsilon }_1\le {\epsilon }_2$, then ${\delta }_f({\epsilon }_1,x_0)\le {\delta }_f({\epsilon }_2,x_0)$.

(v) 

Input translation. For $g\left(x\right)=f(x+a)$ one has ${\delta }_g(\epsilon ,x_0)={\delta }_f(\epsilon ,x_0+a)$.

(vi) 

Lipschitz lower bound. If f is L-Lipschitz on $|x-x_0|<r$[9], then ${\delta }_f(\epsilon ,x_0)\ge min\{r,\epsilon /L\}$; in particular, if f is globally L-Lipschitz, ${\delta }_f(\epsilon ,x_0)\ge \epsilon /L$.

(vii) 

Continuity criterion. f is continuous at $x_0$ iff ${\delta }_f(\epsilon ,x_0)>0$ for every $\epsilon >0$.

(B) Two–function properties.

(viii) 

Composition. If g is continuous at $f\left(x_0\right)$ with modulus η (i.e., $|u-v|<\eta \left(\epsilon \right)\Rightarrow \left|g\right(u)-g(v\left)\right|<\epsilon$), then

$${\delta }_{g\circ f}(\epsilon ,x_0)\ge {\delta }_f(\eta \left(\epsilon \right),x_0).$$

(ix) 

Sum (lower bound). For $f_1,f_2$ and any $\alpha \in [0,1]$,

$${\delta }_{f_1+f_2}(\epsilon ,x_0)\ge min\{{\delta }_{f_1}(\alpha \epsilon ,x_0),{\delta }_{f_2}((1-\alpha )\epsilon ,x_0)\}.$$

(There is no universal matching upper bound; e.g., $f_1=g$, $f_2=-g$.)

(x) 

Linear combination (lower bound). For $c_1,c_2\in \mathbb{R}$ and any $\alpha \in [0,1]$,

$${\delta }_{c_1{f}_1+c_2{f}_2}(\epsilon ,x_0)\ge min\left\{{\delta }_{f_1}\left({\displaystyle \frac{\alpha \epsilon }{|c_1|}},x_0\right),{\delta }_{f_2}\left({\displaystyle \frac{(1-\alpha )\epsilon }{|c_2|}},x_0\right)\right\},$$

with the convention from (ii) when a coefficient is 0.

(xi) 

Product (one convenient bound). If $|f_1|\le A$ on $|x-x_0|<r$, then for any $\epsilon >0$,

$${\delta }_{f_1{f}_2}(\epsilon ,x_0)\ge min\left\{r,{\delta }_{f_2}\left({\displaystyle \frac{\epsilon }{2A}},x_0\right),{\delta }_{f_1}\left({\displaystyle \frac{\epsilon }{2\left(\right|f_2\left(x_0\right)|+1)}},x_0\right)\right\}.$$

Proof. 

(i) ${\omega }_f(x_0;\delta )={\omega }_f(-x_0;\delta )$ for even f (substitute $h\mapsto -h$). (ii) ${\omega }_{cf}=\left|c\right|{\omega }_f$; for $c=0$, $\omega \equiv 0$. (iii) $|f\left(x\right)-f\left(x_0\right)|\le |f\left(x\right)|+|f\left(x_0\right)|\le 2M$. (iv) $\{{\omega }_f\le {\epsilon }_1\}\subseteq \{{\omega }_f\le {\epsilon }_2\}$. (v) $|f(x+a)-f(x_0+a)|=|g\left(x\right)-g\left(x_0\right)|$. (vi) On $|x-x_0|<r$, $|f\left(x\right)-f\left(x_0\right)|\le L|x-x_0|$; take $\delta \le min\{r,\epsilon /L\}$. (vii) Equivalent to the ε–δ definition. (viii) If $|f\left(x\right)-f\left(x_0\right)|<\eta \left(\epsilon \right)$ then $|g\left(f\left(x\right)\right)-g\left(f\left(x_0\right)\right)|<\epsilon$; hence ${\Delta }_f(\eta \left(\epsilon \right),x_0)\subseteq {\Delta }_{g\circ f}(\epsilon ,x_0)$. (ix) By triangle inequality and a split $\alpha \epsilon ,(1-\alpha )\epsilon$; the counterexample $f_1=g$, $f_2=-g$ shows no universal upper bound. (x) Combine (ii) with (ix). (xi) Use $|f_1{f}_2-f_1\left(x_0\right)f_2\left(x_0\right)|\le |f_1|·|f_2-f_2\left(x_0\right)|+|f_2\left(x_0\right)|·|f_1-f_1\left(x_0\right)|$, bound $|f_1|\le A$ on $|x-x_0|<r$, and choose the two tolerances to sum to ε. □

Theorem 1

(Characterization of Radius of Pointwise Continuity). Let $f:\mathbb{R}\to \mathbb{R}$ be a real-valued function with a nonempty set of continuity points $C\left(f\right)\ne ⌀$. For $x_0\in \mathbb{R}$ and $\epsilon >0,$ the following are equivalent:

(i)

f is non-constant.

(ii)

There is $x_0\in C\left(f\right)$ there exists $\epsilon >0$ such that ${\delta }_f(\epsilon ,x_0)<\infty$.

Proof. 

(ii) ⇒ (i). Suppose (ii) holds and, towards a contradiction, assume f is constant: $f\left(x\right)\equiv c_0$. Fix any $x_0\in C\left(f\right)$ (indeed, $C\left(f\right)=\mathbb{R}$) and any $\epsilon >0$. Then $|f\left(x\right)-f\left(x_0\right)|=0<\epsilon$ for all x, so ${\Delta }_f(\epsilon ,x_0)=(0,\infty )$ and thus ${\delta }_f(\epsilon ,x_0)=\infty$, contradicting (ii).

(i) ⇒ (ii). Assume f is non-constant. Fix $x_0\in C\left(f\right)\ne ⌀$. Then there are $a,b\in \mathbb{R}$ with $a\ne b$ and $f\left(a\right)\ne f\left(b\right)$. Set

$$d_a:=|f\left(a\right)-f\left(x_0\right)|,d_b:=|f\left(b\right)-f\left(x_0\right)|,{\epsilon }_b:=\frac{1}{2}\left(d_a+d_b\right).$$

Since $max\{d_a,d_b\}\ge {\epsilon }_b$, at least one of a or b (call it p) satisfies $|f\left(p\right)-f\left(x_0\right)|\ge {\epsilon }_b$. Consequently, any $\delta >max\left\{\right|a-x_0|,|b-x_0\left|\right\}\ge |p-x_0|$ forces the open interval $(x_0-\delta ,x_0+\delta )$ to contain p, and hence to contain a point where $|f\left(x\right)-f\left(x_0\right)|\ge {\epsilon }_b$. By the definition of ${\delta }_f$ we obtain the finite bound

$${\delta }_f({\epsilon }_b,x_0)\le max\left\{\right|a-x_0|,|b-x_0\left|\right\}<\infty .$$

This supplies, for the given $x_0\in C\left(f\right)$, an explicit $\epsilon :={\epsilon }_b>0$ with ${\delta }_f(\epsilon ,x_0)<\infty$. Since $x_0\in C\left(f\right)$ was arbitrary, (ii) follows. □

Corollary 1.

With ${\delta }_f(\epsilon ,x_0)$ defined as above, the following are equivalent:

(i)

f is constant.

(ii)

For every $x_0\in C\left(f\right)$ and every $\epsilon >0$, we have ${\delta }_f(\epsilon ,x_0)=\infty$.

2.2. Radius of Uniform Continuity

In this section we pass from the local picture at a single point to a global one by introducing, for each function f and accuracy level $\epsilon >0$, the set ${\Delta }_f^U\left(\epsilon \right)$ of all radii $\delta >0$ that make the $\epsilon$–$\delta$ condition hold simultaneously for every $x_0\in \mathbb{R}$. The uniform radius of continuity${\delta }_f^U\left(\epsilon \right)$ is then defined as the supremum of this set and measures, in a single number, how far one can move anywhere in the domain without exceeding the prescribed error $\epsilon$ (Figure 2). We show that ${\delta }_f^U$ admits a clean formulation in terms of the uniform oscillation of f[6,8], obtained by taking the supremum of $\left|f\right(x)-f(y\left)\right|$ over all pairs of points at distance less than $\delta$. This viewpoint allows us to derive a collection of general properties of ${\delta }_f^U$ and to compare it directly with the pointwise radius ${\delta }_f(\epsilon ,x_0)$, for instance by bounding the former in terms of the infimum of the latter over all base points. In particular, whenever f is uniformly continuous, the uniform radius is strictly positive for each $\epsilon >0$, while for merely pointwise continuous functions ${\delta }_f^U\left(\epsilon \right)$ may degenerate to zero even though each ${\delta }_f(\epsilon ,x_0)$ remains positive. Finally, we obtain a characterization analogous to the pointwise case: the pattern of finiteness or infiniteness of ${\delta }_f^U\left(\epsilon \right)$ completely determines whether f is constant or non-constant, so that the uniform radius encodes global regularity information in a quantitative form.

Definition 2

(Radius of Uniform Continuity). Let $f:\mathbb{R}\to \mathbb{R}$ be a real-valued function with a nonempty set of continuity points $C\left(f\right)\ne ⌀$. For $\epsilon >0$ define the uniform admissible radius as:

$$\begin{array}{ccc}\hfill {\Delta }_f^{\mathrm{U}}\left(\epsilon \right)& :=& \left\{\delta >0:\forall x\in \mathbb{R}\forall y\in \mathbb{R}\left(|x-y|<\delta \Rightarrow \left|f\right(x)-f(y\left)\right|<\epsilon \right)\right\},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\delta }_f^{\mathrm{U}}\left(\epsilon \right)& :=& sup{\Delta }_f^{\mathrm{U}}\left(\epsilon \right).\hfill \end{array}$$

(8)

Remark 2

(Shifted-h formulation of ${\Delta }_f^U$ and ${\delta }_f^U$). For the uniform case, introduce theuniform local oscillation

$$\begin{array}{c}\hfill {\omega }_f^U\left(\delta \right):=\underset{x\in \mathbb{R}}{sup}\underset{\left|h\right|<\delta }{sup}|f(x+h)-f\left(x\right)|.\end{array}$$

(9)

Then:

$$\begin{array}{ccc}\hfill {\Delta }_f^U\left(\epsilon \right)& :=& \{\delta >0:{\omega }_f^U\left(\delta \right)<\epsilon \},\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\delta }_f^U\left(\epsilon \right)& :=& sup\{\delta >0:{\omega }_f^U\left(\delta \right)<\epsilon \}.\hfill \end{array}$$

(11)

Proposition 2

(Properties of Radius of Uniform Continuity). Let $f:\mathbb{R}\to \mathbb{R}$. Then, for $\epsilon >0$ the radius of uniform continuity ${\delta }_f^U\left(\epsilon \right)$ has the following properties:

(A) Single–function properties.

(i) 

Scaling. If $c\ne 0$, then ${\delta }_{cf}^U\left(\epsilon \right)={\delta }_f^U(\epsilon /|c\left|\right)$; if $c=0$, then ${\delta }_{0·f}^U\left(\epsilon \right)=+\infty$.

(ii) 

Monotonicity in $\epsilon$.If $0<{\epsilon }_1\le {\epsilon }_2$, then ${\delta }_f^U\left({\epsilon }_1\right)\le {\delta }_f^U\left({\epsilon }_2\right)$.

(iii) 

Translation/reflection invariance. For $g\left(x\right)=f(x+a)$ or $g\left(x\right)=f(-x)$, we have ${\delta }_g^U\left(\epsilon \right)={\delta }_f^U\left(\epsilon \right)$.

(iv) 

Large–$\epsilon$ for bounded f. If $\left|f\right|\le M$ everywhere and $\epsilon \ge 2M$, then ${\delta }_f^U\left(\epsilon \right)=+\infty$ (sharp in general).

(v) 

Lipschitz lower bound. If f is globally L-Lipschitz, then ${\delta }_f^U\left(\epsilon \right)\ge \epsilon /L$.

(vi) 

Uniform continuity criterion. f is uniformly continuous on $\mathbb{R}$ iff ${\delta }_f^U\left(\epsilon \right)>0$ for every $\epsilon >0$.

(vii) 

Link to pointwise radii. ${\displaystyle {\delta }_f^U\left(\epsilon \right)\le \underset{x_0\in \mathbb{R}}{inf}{\delta }_f(\epsilon ,x_0)}$.

(B) Two–function properties.

(viii) 

Composition. If g admits a global modulus η with $|u-v|<\eta \left(\epsilon \right)\Rightarrow \left|g\right(u)-g(v\left)\right|<\epsilon$, then

$${\delta }_{g\circ f}^U\left(\epsilon \right)\ge {\delta }_f^U\left(\eta \left(\epsilon \right)\right).$$

(ix) 

Sum (lower bound). For any $\alpha \in [0,1]$,

$${\delta }_{f_1+f_2}^U\left(\epsilon \right)\ge min\left\{{\delta }_{f_1}^U\left(\alpha \epsilon \right),{\delta }_{f_2}^U\left((1-\alpha )\epsilon \right)\right\}.$$

(There is no universal matching upper bound; e.g., $f_1=g$, $f_2=-g$.)

(x) 

Linear combination (lower bound). For $c_1,c_2\in \mathbb{R}$ and any $\alpha \in [0,1]$,

$${\delta }_{c_1{f}_1+c_2{f}_2}^U\left(\epsilon \right)\ge min\left\{{\delta }_{f_1}^U\left({\displaystyle \frac{\alpha \epsilon }{|c_1|}}\right),{\delta }_{f_2}^U\left({\displaystyle \frac{(1-\alpha )\epsilon }{|c_2|}}\right)\right\},$$

with the convention in (i) if a coefficient is 0.

(xi) 

Product (global bounds). If $|f_1|\le A$ and $|f_2|\le B$ on $\mathbb{R}$, then for any $\epsilon >0$,

$${\delta }_{f_1{f}_2}^U\left(\epsilon \right)\ge min\left\{{\delta }_{f_2}^U\left({\displaystyle \frac{\epsilon }{2A}}\right),{\delta }_{f_1}^U\left({\displaystyle \frac{\epsilon }{2B}}\right)\right\}.$$

Proof. 

(i) ${\omega }_{cf}^U\left(\delta \right)=\left|c\right|{\omega }_f^U\left(\delta \right)$. If $c=0$, ${\omega }^U\equiv 0$. (ii) Inclusion of level sets $\{{\omega }_f^U\le {\epsilon }_1\}\subseteq \{{\omega }_f^U\le {\epsilon }_2\}$. (iii) Both translations and $x\mapsto -x$ leave ${\omega }^U$ unchanged: $\left|f\right(x+a+h)-f(x+a\left)\right|$ and $\left|f\right(-x-h)-f(-x\left)\right|$. (iv) For all $x,y$, $\left|f\right(x)-f(y\left)\right|\le \left|f\right(x\left)\right|+\left|f\right(y\left)\right|\le 2M$. (v) Global Lipschitz gives $\left|f\right(x)-f(y\left)\right|\le L|x-y|$; take $\delta \le \epsilon /L$. (vi) This is exactly the ε–δ definition of uniform continuity. (vii) If $\delta \in {\Delta }_f^U\left(\epsilon \right)$ then, fixing any $x_0$, $|x-x_0|<\delta \Rightarrow |f\left(x\right)-f\left(x_0\right)|<\epsilon$, hence $\delta \in {\Delta }_f(\epsilon ,x_0)$; take sups and then the infimum over $x_0$. (viii) If $\left|f\right(x)-f(y\left)\right|<\eta \left(\epsilon \right)$ then $\left|g\right(f\left(x\right))-g(f\left(y\right)\left)\right|<\epsilon$; thus ${\Delta }_f^U\left(\eta \left(\epsilon \right)\right)\subseteq {\Delta }_{g\circ f}^U\left(\epsilon \right)$. (ix) Triangle inequality with split $\alpha \epsilon ,(1-\alpha )\epsilon$. (x) Apply (i) to $c_1{f}_1$ and $c_2{f}_2$ and then (ix). (xi) Using $|f_1{f}_2\left(x\right)-f_1{f}_2\left(y\right)|\le |f_1\left(x\right)\left|\right|f_2\left(x\right)-f_2\left(y\right)|+|f_2\left(y\right)\left|\right|f_1\left(x\right)-f_1\left(y\right)|\le A{\Delta }_2+B{\Delta }_1$, pick ${\Delta }_2<\epsilon /\left(2A\right)$ and ${\Delta }_1<\epsilon /\left(2B\right)$ and invoke the uniform radii of $f_2$ and $f_1$, respectively. □

Theorem 2

(Characterization of Radius of Uniform Continuity ). Let $f:\mathbb{R}\to \mathbb{R}$ be a real-valued function with a nonempty set of continuity points $C\left(f\right)\ne ⌀$. For $\epsilon >0$ the following are equivalent:

(i)

f is non-constant ⟺

(ii)

There exists $\epsilon >0$, ${\delta }_f^{\mathrm{U}}\left(\epsilon \right)<\infty$.

Proof. 

If f is constant, then $\left|f\right(x)-f(y\left)\right|=0<\epsilon$ for all $x,y$, so ${\Delta }_f^{\mathrm{U}}\left(\epsilon \right)=(0,\infty )$ and ${\delta }_f^{\mathrm{U}}\left(\epsilon \right)=\infty$.

Conversely, suppose f is non-constant, pick $a\ne b$ with $f\left(a\right)\ne f\left(b\right)$ and set ${\epsilon }_0:=\frac{1}{2}|f\left(a\right)-f\left(b\right)|>0$. Then for any $\delta >|a-b|$, the pair $(x,y)=(a,b)$ satisfies $|x-y|<\delta$ but $|f\left(x\right)-f\left(y\right)|\ge {\epsilon }_0$, so $\delta \notin {\Delta }_f^{\mathrm{U}}\left({\epsilon }_0\right)$. Hence

$${\delta }_f^{\mathrm{U}}\left({\epsilon }_0\right)\le |a-b|<\infty .$$

This mirrors the pointwise result with ${\delta }_f(\epsilon ,x_0)$ and shows the same finite-radius phenomenon in the uniform setting. □

Corollary 2.

With ${\delta }_f^U\left(\epsilon \right)$ defined as above, the following are equivalent:

(i)

f is constant.

(ii)

For every $\epsilon >0$, we have ${\delta }_f\left(\epsilon \right)=\infty$.

3. Examples and Explicit Computations

In this section we illustrate the general theory by working out two concrete families of functions in detail. First, in Section 3.1 we study the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ and compute an explicit closed-form expression for the pointwise radius of continuity ${\delta }_f(\epsilon ,x_0)$ in terms of $\epsilon$, $x_0$, and the coefficients $a,b,c$. This formula allows us to analyze how the radius behaves as the base point $x_0$ moves along the real line, showing in particular that it decays at infinity and attains its maximal value at the vertex $x_0=-b/\left(2a\right)$. We also identify a simple symmetry condition on the coefficients under which the radius becomes an even function of $x_0$. Second, in Section 3.2 we turn to a probabilistic example and consider the normal density with parameters $\mu$ and ${\sigma }^2$. Using an explicit Lipschitz constant for this density, we derive a closed-form expression for the uniform radius of continuity ${\delta }_f^U\left(\epsilon \right)$, which exhibits a piecewise behaviour in $\epsilon$: linear growth up to a threshold followed by saturation at infinity. The resulting formula makes transparent how the uniform radius depends on the scale parameter $\sigma$, and thereby demonstrates how our radius-of-continuity framework captures both geometric and probabilistic features of a function in a quantitative way.

3.1. Radius of Pointwise Continuity of the Quadratic Polynomial

Example 1.

Let $f\left(x\right)=a{x}^2+bx+c(a\ne 0)$ be the quadratic polynomial. We calculate ${\delta }_f(\epsilon ,x_0)$ in the following three steps:

Step 1 — Increment at $x_0$.

$$\begin{array}{ccc}\hfill f(x_0+h)-f\left(x_0\right)& =& a\left[{(x_0+h)}^2-x_0^2\right]+bh\hfill \\ & =& a(2x_{0}h+h^2)+bh\hfill \\ & =& a{h}^2+(2a{x}_0+b)h.\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill B& :=& 2a{x}_0+b.\hfill \end{array}$$

(13)

Step 2 — Exact oscillation on $\left|h\right|<\delta$.

$$\begin{array}{ccc}\hfill \underset{\left|h\right|\le \delta }{sup}|a{h}^2+Bh|& =& max\left\{|a{\delta }^2+B\delta |,|a{\delta }^2-B\delta |\right\}\hfill \\ & =& \left|a\right|{\delta }^2+\left|B\right|\delta .\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\omega }_f(x_0;\delta )& =& \left|a\right|{\delta }^2+|2a{x}_0+b|\delta .\hfill \end{array}$$

(15)

Step 3 — Solve ${\omega }_f(x_0;\delta )<\epsilon$.

$$\begin{array}{ccc}\hfill {\delta }_f(\epsilon ,x_0)& =& sup\left\{\delta >0:\left|a\right|{\delta }^2+|2a{x}_0+b|\delta <\epsilon \right\}\hfill \\ & =& sup\left\{\delta >0:\left|a\right|{\delta }^2+|2a{x}_0+b|\delta -\epsilon \le 0\right\}\hfill \\ & =& {\displaystyle \frac{-|2a{x}_0+b|+\sqrt{|2a{x}_0+b{|}^2+4\left|a\right|\epsilon }}{2\left|a\right|}}\hfill \\ & =& {\displaystyle \frac{2\epsilon }{|2a{x}_0+b|+\sqrt{|2a{x}_0+b{|}^2+4\left|a\right|\epsilon }}}.\hfill \end{array}$$

(16)

Behavior and Evaluations

We have:

1.

Asymptotic Equivalency:

$$\begin{array}{ccc}\hfill {\delta }_f(\epsilon ,x_0)& \sim & {\displaystyle \frac{\epsilon }{2\left|a\right||x_0|}}:x_0\to \pm \infty .\hfill \end{array}$$

(17)

2.

Left/Right Branch:

$$\begin{array}{ccc}\hfill {\displaystyle \underset{x_0\to \pm \infty }{lim}{\delta }_f(\epsilon ,x_0)}& =& 0.\hfill \end{array}$$

(18)

3.

Vertex:

$$\begin{array}{ccc}\hfill {\delta }_f\left(\epsilon ,-{\displaystyle \frac{b}{2a}}\right)& =& \sqrt{{\displaystyle \frac{\epsilon }{\left|a\right|}}}.\hfill \end{array}$$

(19)

4.

Symmetry Condition:

$$\begin{array}{ccc}\hfill \forall \epsilon \in {\mathbb{R}}^{+}\forall x_0\in \mathbb{R}:{\delta }_f(\epsilon ,-x_0)={\delta }_f(\epsilon ,x_0)& \iff & b=0.\hfill \end{array}$$

(20)

5.

Figure 3 presents the radius of pointwise continuity for the quadratic polynomial function.

3.2. Radius of Uniform Continuity of the General Normal Probability Density Function

Example 2. 

Let $f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}exp\left(-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}\right)$ be the probability density of the normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$. We calculate ${\delta }_f^U\left(\epsilon \right)$ as follows:

Step 1 (Function and derivative). 

$$\begin{array}{ccc}\hfill f^{\prime }\left(x\right)& =& -{\displaystyle \frac{x-\mu }{{\sigma }^2}}f\left(x\right)=-{\displaystyle \frac{x-\mu }{\sqrt{2\pi }{\sigma }^3}}exp\left(-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}\right).\hfill \end{array}$$

(21)

Step 2 (Global Lipschitz constant). Let $L:={sup}_{x\in \mathbb{R}}\left|f^{\prime }\left(x\right)\right|$. Set $t=(x-\mu )/\sigma$. For $t\ge 0$, $g\left(t\right):=t{e}^{-t^2/2}$ has

$$g^{\prime }\left(t\right)=e^{-t^2/2}(1-t^2)=0\iff t=1,$$

and g is maximized at $t=1$ with $g\left(1\right)=e^{-1/2}$. Hence

$$\begin{array}{ccc}\hfill L& =& {\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }^2}}{e}^{-1/2}={\displaystyle \frac{1}{\sqrt{2\pi e}{\sigma }^2}}.\hfill \end{array}$$

(22)

Step 3 (Uniform oscillation upper bound). By the mean value theorem, for all $x\in \mathbb{R}$ and $\left|h\right|<\delta$,

$$\begin{array}{ccc}\hfill \left|f\right(x+h)-f(x\left)\right|& \le & L\left|h\right|\le L\delta .\hfill \end{array}$$

Taking the sup over x and $\left|h\right|<\delta$ gives

$$\begin{array}{ccc}\hfill {\omega }_f^U\left(\delta \right)& \le & L\delta .\hfill \end{array}$$

(23)

Step 4 (Sharpness—matching lower bound). Choosing x near $\mu \pm \sigma$ (where $|f^{\prime }|$ is maximal) and $\left|h\right|\downarrow 0$ with the sign of $f^{\prime }$,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\left|f\right(x+h)-f(x\left)\right|}{\left|h\right|}}& ⟶& |f^{\prime }\left(x\right)|\approx L,\hfill \end{array}$$

so for any $\eta >0$ and small enough δ,

$${\omega }_f^U\left(\delta \right)\ge (L-\eta )\delta .$$

Together with (23),

$$\begin{array}{ccc}\hfill {\omega }_f^U\left(\delta \right)& =& L\delta (\delta >0).\hfill \end{array}$$

(24)

Step 5 (Saturation ceiling). $max\left(f\right)-min\left(f\right)=f\left(\mu \right)-0=1/\left(\sqrt{2\pi }\sigma \right)$. Thus

$$\begin{array}{ccc}\hfill {\omega }_f^U\left(\delta \right)& =& min\left\{L\delta ,{\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}\right\}.\hfill \end{array}$$

(25)

Step 6 (Solve for $\delta$).By definition and (25),

$$\begin{array}{ccc}\hfill {\delta }_f^U\left(\epsilon \right)& =& sup\{\delta >0:{\omega }_f^U\left(\delta \right)\le \epsilon \}\hfill \\ & =& sup\left\{\delta >0:min\{L\delta ,\frac{1}{\sqrt{2\pi }\sigma }\}\le \epsilon \right\}\hfill \\ & =& \left\{\begin{array}{cc}\left(\sqrt{2\pi e}{\sigma }^2\right)\epsilon ,\hfill & 0<\epsilon <{\displaystyle \frac{1}{\sqrt{2\pi }\sigma }},\hfill \\ +\infty ,\hfill & \epsilon \ge {\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}.\hfill \end{array}\right.\hfill \end{array}$$

(26)

Behavior and Evaluations

We have:

1.

Asymptotic (small ε):

$$\begin{array}{ccc}\hfill {\delta }_f^U\left(\epsilon \right)& \sim & \sqrt{2\pi e}{\sigma }^2\epsilon :\epsilon \downarrow 0.\hfill \end{array}$$

(27)

2.

Parameter dependence:

The radius is independent of μ. For fixed $\epsilon <{\epsilon }_{*}=1/\left(\sqrt{2\pi }\sigma \right)$, ${\delta }_f^U\left(\epsilon \right)$ grows quadratically in σ ($\propto {\sigma }^2$). The threshold ${\epsilon }_{*}=1/\left(\sqrt{2\pi }\sigma \right)$ decreases like $1/\sigma$, and the break radius ${\delta }_{*}=\sigma \sqrt{e}$ increaseslinearlyin σ.

3.

Monotonicity and jump:

${\delta }_f^U$ is strictly increasing and linear on $(0,{\epsilon }_{*})$ with slope $\sqrt{2\pi e}{\sigma }^2$. It has a jump at $\epsilon ={\epsilon }_{*}$:

$$\begin{array}{c}\hfill \underset{\epsilon \uparrow {\epsilon }_{*}}{lim}{\delta }_f^U\left(\epsilon \right)={\delta }_{*}=\sigma \sqrt{e},{\delta }_f^U\left({\epsilon }_{*}\right)=+\infty .\end{array}$$

(28)

4.

Scaling identity:

If $f_{(\mu ,\sigma )}$ is the pdf for $\mathcal{N}(\mu ,{\sigma }^2)$ distribution and $f_{(0,1)}$ is pdf for the standard normal distribution, then for $0<\epsilon <{\epsilon }_{*}$:

$$\begin{array}{c}\hfill {\delta }_{f_{(\mu ,\sigma )}}^U\left(\epsilon \right)={\sigma }^2{\delta }_{f_{(0,1)}}^U\left(\sigma \epsilon \right).\end{array}$$

(29)

5.

Figure 4 presents the radius of uniform continuity for the normal probability density function.