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Suppose that a continuous function $f: \mathbb{R} \mapsto \mathbb{R}$ is non-smooth, but has a single maximum on the interval $[a,b]$.

Inspired by Riemann integrals, I started to consider

$$\lim_{n \rightarrow \infty}\max \left( f(a), f\left(\frac{b-a}{n} + a \right), f\left(\frac{2}{n}(b-a) + a \right), \ldots, f\left(\frac{n-1}{n}(b-a)+a\right), f(b) \right)$$

with the desire to convert this into a limit of a series iteratively using the relation $$\max(x,y) = \frac{x+y+|x-y|}{2}.$$

But I find that it doesn't immediately give a nice series. For the case $n=2$ I get

$$\max \left( f(a), f\left(\frac{b-a}{2} + a \right), f(b) \right)$$

$$=$$

$$\frac{\frac{f(a) + f\left(\frac{b-a}{2} + a \right) + \left|f(a) -f\left(\frac{b-a}{2} + a \right) \right|}{2} + f(b) + \left|\frac{f(a) + f\left(\frac{b-a}{2} + a \right) + \left|f(a) -f\left(\frac{b-a}{2} + a \right) \right|}{2} - f(b) \right|}{2}$$

which I don't know how to proceed from. Is there a way to proceed with setting up the limit so it can be evaluated?

Galen
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  • Hmm, maybe I can use the triangle inequality could simplify the terms somewhat by looking at an upper bound for the maximum for any given finite sequence. – Galen Apr 17 '23 at 05:07
  • Using the triangle inequality for the $n=2$ case: $$\max \left( f(a), f\left(\frac{b-a}{2} + a \right), f(b) \right) \leq \frac{1}{4}\left[f(a) + f\left(\frac{b-a}{2} + a \right)+ 2f(b) + 3\left|f(a)\right| + 3\left|f\left(\frac{b-a}{2} + a \right)\right| + 2\left|f(b) \right|\right]$$ – Galen Apr 17 '23 at 05:20
  • The MCT implies that the sequence of maxima will converge. – Galen Apr 17 '23 at 05:38

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