Let $f$ be an increasing function on $[0,1]$ . Let $F(x)= \int_0^x f(t) dt$. I want to show integrals are well defined. My attempts:
$f$ is bounded, $f(0)\leq f(x)\leq f(1)$.
$f$ may only have a countable set of jump since it's increasing.
How can I show that between jumps I have continuity?