A uniformly continuous function is always continuous. But the converse is not true. For example, f(x)= 1/x on (0, 2). In this post, we will provide an example which is continuous but not uniformly continuous. Before we do that let us recall their definitions.

**Continuous Function:** A function f(x) is said to be continuous at x=a if for every ε>0 there exists a δ>0 (depending on ε and the point x=a, so we write δ:=δ(ε, a)) such that

|f(x)-f(a)| < ε for all |x-a| < δ.

**Uniform Continuous Function:** A function f(x) is said to be uniformly continuous at x=a if for every ε>0 there exists a δ>0 (not depend on ε and the point x=a) such that for any two points x_{1} and x_{2} with |x_{1} – x_{2}| < δ we have

|f(x_{1})-f(x_{2})| < ε.

To show continuity does not imply uniform continuity, let us consider the following function.

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## An Example

### f(x) = 1/x is continuous on (0,2) but not uniformly continuous

Let

$f(x)=\dfrac{1}{x}$ on (0, 2).

See that f(x) is continuous on (0,2) as the only point of discontinuity is x=0. We now show that f(x) is not uniformly continuous.

Note that if f(x) = 1/x was uniformly continuous on (0, 2) then we would have that ∀ ε>0 ∃ a positive δ such that

|f(x)-f(y)| < ε whenever |x-y| < δ.

That is,

$|\dfrac{1}{x}-\dfrac{1}{y}|$ < ε whenever |x-y| < δ …(∗)

Take ε = 1 and x:= min{δ, ε} = min{δ, 1} where δ>0 is arbitrary. Also, consider $y=\dfrac{x}{2}$. Then

$|x-y|=|x-\dfrac{x}{2}| =\dfrac{x}{2}<\delta$.

But,

$|\dfrac{1}{x}-\dfrac{1}{y}|$ = $|\dfrac{1}{x}-\dfrac{2}{x}|$ = $|\dfrac{1}{x}|$ ≥ 1 = ε.

This contradicts the fact (∗) above, showing that f(x) is not uniformly continuous on(0, 2). So f(x) = 1/x is continuous on (0, 2) but not uniformly continuous.

## Another Example

Consider the function

$f(x)=\dfrac{1}{x}$ on (0, 1].

It is a continuous function. To show it is a uniformly continuous, for every pair of sequences {a_{n}} and {b_{n}} in (0, 1] with lim |a_{n} – b_{n}| = 0 the following holds:

lim |f(a_{n}) – f(b_{n})| = 0.

Take a_{n} = 1/n and b_{n} = 1/2n. Then

lim |a_{n} – b_{n}| = lim | 1/n – 1/2n | = 0.

But, lim |f(a_{n}) – f(b_{n})| = lim n ≠ 0.

So f(x) = 1/x on (0, 1] is not uniformly continuous but it is a continuous function.

You Can Read:

- Intermediate Value Theorem
- Fixed Point Theorem
- Completeness Axiom of Real Numbers
- Supremum & Infimum of a Set
- Archimedean Property

## FAQ

**Q1: Give an example of a continuous function that is not uniformly continuous.**

Answer: The function f(x) = 1/x defined on (0, 2) is continuous but not uniformly continuous.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.