Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions

Let $[a,b]$ be a closed real interval, $\mathbb R$ be the set of real numbers and \(f:[a,b]\mapsto \mathbb R\) a continuous function. Then for every \(\epsilon > 0\) there exist some step functions \(\phi:[a,b]\mapsto\mathbb R\) and \(\psi:[a,b]\mapsto\mathbb R\) with

$(i)$ $\phi(x) \le f(x)\le \psi(x)$ for all $x\in[a,b],$ and

$(ii)$ $|\phi(x)-\psi(x)|\le\epsilon$ for all $x\in[a,b].$


The following interactive figure demonstrates the idea of this lemma for the close interval $[-3,3]$, some continuous function $f$ and the step function \(\phi\) green and \(\psi\) red. You can drag the red points to change the curvature of the function and use the slider to change the number of steps of the step functions.

Proofs: 1

Proofs: 1

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