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Let $f: [-\pi, \pi] \to \mathbb{R}$ be a bounded and monotonic function. Let $M > 0$ such that $|f(x)| \leq M$ for all $x \in [-\pi, \pi]$. Can $f$ be approximated (in $L^{\infty}$-sense) by a function in a form $$\sum_{k=1}^N\alpha_k\mathbb{1}_{[a_k, a_{k+1}]}(x)$$ with $-\pi = a_1 < a_2 < \cdots< a_{N+1} = \pi$ and $|\alpha_j| \leq M$ for all $j$?
When I draw a picture of $f$, then surely we should be able to do it. But how do you write it formally? I believe you don't want the step from $f(a_1)$ to $f(a_2)$to be too large, for instance. So, we should choose $a_2$ such that $f(a_2) < \varepsilon + f(a_1)$ where $\varepsilon > 0$ is small? Any ideas or hints would be appreciated

Vicky
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  • Do $a_1, a_2$ depend on $N$? I mean $a_1, a_2$ can be changed when $N$ tends to $\infty$? – Leonard Neon Jan 08 '21 at 19:47
  • Partition $[-M,M]$ into subintervals of equal length, and look at the inverse image of these subintervals under $f$. – David Mitra Jan 08 '21 at 19:54
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    A monotone function is measurable, and measurable functions can be approximated by step-functions. Moreover monotone functions are differentiable a.e. – Matías Ures Jan 08 '21 at 19:54

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