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?