If $f,g: [0, 1] \to \mathbb{R} $ are increasing and non-negative functions, show that the function $h(x, y) = f(x)g(y)$ is integrable over $[0, 1]^2$

Question

In the book of Analysis on Manifolds at page 90 question 4, it is asked that

We say $f:[0,1] \to \mathbb{R}$ is increing if $f(x_1)<f(x_2)$ whenever $x_1<x_2$. If $f, g : (0, 1] \to \mathbb{R} $ are increing and non-negative, show that the function $h(x, y) = f(x)g(y)$ is integrable over $[0, 1]^2$.

and the same question has been asked in here; however, how do we know the values $f(1)$ and $g(1)$ are finite ? I mean there is nothing that assumes the functions are bounded in the statement of the question as far as I can see.

Answer 1

We have $f, g : (0, 1] \to \mathbb{R}$, hence $f((0,1]) \subseteq \mathbb R$ and $g((0,1]) \subseteq \mathbb R$ , therefore $f(1),g(1) \in \mathbb R$.