Why is f bounded? I am looking at the exercise: Let $f:[0,1] \to \mathbb{R}$ integrable such that $|f(x)| \leq \int_0^x |f(t)|dt, \forall x \in [0,1]$.
Show that $f=0$.
At the solution,it is taken that $f$ is bounded. Why??Because of the fact that $f$ is integrable?

The function $F(x)=\int_0^x |f(t)|dt$ is continuous and is defined on the compact set $[0,1]$; hence, $F$ is bounded. Since $|f(x)|\leq F(x)$ for each $x\in[0,1]$, $f$ is also bounded.
