discretization of continuous function and its maximum

Question

This is likely to be a silly question, but I'm not completely sure about its answer.

Let $f:[0,1]\rightarrow [0,1]$ be a continuous function that attains its maximum at some $x^*\in (0,1)$. Now suppose that I take a grid $[x_1,\ldots,x^*,\ldots,x_n]$ where $x_1=0<x_2<x_3<\cdots <x_n=1$. Then, does it remain true that $f(x^*)\geq f(x)$ for all $x_i$ in the grid? Is that simple or I am missing something here?

Thanks!

Answer 1

If $f(x^*)$ is the maximum of the function over $[0,1]$, then $f(x^*)\geq f(x)$ for all $x\in[0,1]$. In particular, $f(x^∗)≥f(x_i)$ for $i=1,…,n$.