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Definition: Uniformly Continuous Functions (Real Case)
Let \(D\subset\mathbb R\) be a subset of real numbers $\mathbb R$ and let $f:D\mapsto \mathbb R$ be a function $f$ is called uniformly continuous, if and only if for every $\epsilon > 0$ there is a $\delta > 0$ such that
$$f(x)f(y) < \epsilon$$
for all $x,y\in D$ with $$xy < \delta.$$
Table of Contents
Corollaries: 1
 Proposition: Not all Continuous Functions are also Uniformly Continuous
 Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions
 Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous
Mentioned in:
Corollaries: 1
Parts: 2
Proofs: 3 4
Propositions: 5
Theorems: 6
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983