Definition:Uniform Continuity/Real Function

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Definition

Let $I \subseteq \R$ be a real interval.

A real function $f: I \to \R$ is said to be uniformly continuous on $I$ if and only if:

for every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds:
for every $x, y \in I$ such that $\size {x - y} < \delta$ it happens that $\size {\map f x - \map f y} < \epsilon$.


Formally: $f: I \to \R$ is uniformly continuous if and only if the following property holds:

$\forall \epsilon > 0: \exists \delta > 0: \paren {x, y \in I, \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon}$



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It can be seen that this says exactly the same thing as the definition for metric spaces if $\R$ is considered a metric space under the Euclidean metric.


Examples

Square Function

Let $S$ be the open interval $S = \openint 0 1$.

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x^2$

Then $f$ is uniformly continuous on $S$.


Also see

  • Results about uniformly continuous real functions can be found here.


Sources

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