TITLE: Prove that this continuous function?
QUESTION [0 upvotes]: Let f be continuous on $\left [ a,b \right ]$ , and a$<$x$<$b.
Prove that:$$\lim_{h\rightarrow 0}\frac{1}{h}\int_{a}^{x}\left ( f\left ( t+h \right )-f\left ( t \right ) \right )dt=f\left(x \right )-f\left(a\right)$$
I have tried Mean Value Theorm to solve it, but it didn't work. And I've been stuck on this for a while without any idea. Is it possible for someone to help me? Or give a hint.
Thanks!

REPLY [0 votes]: Hint: Change of variables in $\int_a^x f(t+h)\; dt$.
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