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 $$|x-y| < \delta.$$

Corollaries: 1

  1. Proposition: Not all Continuous Functions are also Uniformly Continuous
  2. Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions
  3. Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous

Corollaries: 1
Parts: 2
Proofs: 3 4
Propositions: 5
Theorems: 6

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983