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 $x\in[a,b].$

Example

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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References

Bibliography

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