I am reading a notes in which following is written:
Let $f$ be continuous periodic function defined on $ \mathbb R$ with period 2l. Furthermore assume that $f’$ is piecewise continuous then $\int_ l^x f’(t) dt= f(x) $
I think the author is applying Fundamental Theorem of Calculus but $f’$ is just piecewise continuous and I know that Fundamental theorem of calculus is not valid for piecewise continuous function.Could you please help me verifying the above claim?
I tend to suspect that what's actually written in the notes is not the same as what's written above. For example, the integral is actually $f(x)-f(l)$, not $f(x)$. Also it's assumed that $f$ is periodic, but this is totally irrelevant to the conclusion.
If $f$ is continuous, $f$ is differentiable except at finitely many points, and $f'$ is piecewise continuous then it's certainly true that $$f(y)-f(x)=\int_x^yf'(t)\,dt.$$
Suppose for example that $f'$ is continuous on $[x,a)$, continuous on $(a,y]$, and has one-sided limits at $a$. (Note that the existence of one-sided limits is part of the definition of "piecewise continuous".) For small $\delta>0$ we have $$(\int_x^{a-\delta}+\int_{a+\delta}^y)f'(t)\,dt=f(y)-f(a+\delta)+f(a-\delta)-f(x).$$Now let $\delta\to0$: The fact that $f'$ has one-sided limits at $a$ shows that $$(\int_x^{a-\delta}+\int_{a+\delta}^y)f'(t)\,dt\to\int_x^y f'(t)\,dt,$$while the continuity of $f$ shows that $$-f(a+\delta)+f(a-\delta)\to0.$$
Similarly if $f'$ has more than one (but only finitely many) points of discontinuity.