Recall that the first fundamental theorem of calculus says that if $f$ is continuous real values function defined on $[a,b]$. Then the function defined by $F(x)= \int_ a^x f(t) dt$ is differentiable and $F’(x)= f(x) $
Does the same result hold if $f$ is merely piecewise continuous instead of continuous?
Edit: The above question is motivated from a set of notes. The exact statement written in notes is: Let $f$ be continuous on $ \mathbb R$ with period 2l, then $\int_ l^x f’(t) dt= f(x) dx$
P.S: I don’t know about measure theory so please avoid to use it.