Is $g\circ f$ integrable if $f$ is increasing and $g$ is integrable

Question

Let $f:[a, b] \to \Bbb R$ be increasing and $g:f[a, b] \to \Bbb R$ be integrable. Is $g\circ f$ integrable on $[a, b]$?

It seems integrable to me since $g$ is integrable on $[f(a),f(b)]$ since for any partition $P'$ of $[a,b]$, we can choose partition $P''$ of $[f(a),f(b)]$ such that $$U(g\circ f,P')−L(g\circ f,P')≤U(g,P'')−L(g,P'')$$ Is this reasoning correct?

Answer 1

$$f:[0,1]\to[0,1] ,\quad f(x) = x^2 $$ $$ g: [0,1] \to \mathbb R ,\quad g = \begin{cases} \frac{1}{\sqrt x} & x\neq 0\\ 0 & x=0 \end{cases} $$

Notice that $f$ is increasing, $g$ is integrable, while $g\circ f$ is not integrable.