A function like a definite integral on each of the closed subintervals in the real line.

Hello, my question is related to this earlier question
Let f be integrable in [0,1] and assume that f is differentiable in [0,1].
Prove that $f’$ is a difference of integrals in each of the closed subintervals $I_n=[a_n, b_n]$ of $[0,1]$
Let $I=[a,b]$ and $f_i=f$ if $ale ile b$
Then use the assumptions to show that $f_1=f_2$
$f_1$ and $f_2$ are integrable in $[a,b]$ and have continuous differentials $[f_1′]$ and $[f_2′]$ in $[a,b]$ which are Lipschitz functions. Then the mean value formula tells us that $f_1′-f_2′ = $ so $f_1’=f_2’$ in $[a,b]$ because both are defined and both are continuous. Thus they are equal everywhere in [0,1]. The $f’$ is integrable in [0,1] and also on the subintervals $[a_n,b_n]$. It follows that $int_a^b f’dx= sum_n int_{a_n}^{b_n} f’dx = sum_n (b_n-a_n) $


If $f$ is Riemann integrable and $f$ is integrable on $I$, then $f$ is integrable on $I$.

This implies


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