supremum can be move to inside function? when function is strictly increasing?

Question

I want to know if $\sup_{x\in X} f(x) = f(\sup_{x\in X}x)$ where $X$ is a compact space, and $f$ is a strictly increasing function: $\frac{df}{dx} >0$ for all $x$.

Answer 1

I suppose that $X \subseteq \mathbb R$ (otherwise $\sup_{x\in X}x$ makes no sense !)

Let $x_0:= \sup_{x\in X}x$. Since $X$ is compact, we have $x_0 =\max X$. $f$ is increasing, then we have

$f(t) \le f(x_0)$ for all $t \in X$. Therefore

$\max f(X)=f(x_0)$.