Show that this function is in $L_1$

Question

Let's say that function $f\in[0,T]$ is in a class $C$ when $$ f(x) = \frac{g(x)}{x^\alpha} $$ where $0\leq \alpha<1$ and $g(x)$ is continuous on [0,T].

Then, it is trivial to show $C \subset L_1[0,T] $.

My question is that converse is also correct? (that is, $C=L_1[0,T]$)

If so, how to prove it?

If not, what is one of counterexamples?

Thank you!

Answer 1

This is not true. $L^1(0,T)$ contains discontinuous functions.