Suppose $f(x)$ is integrable on an open interval containing $[a,b]$ and is continuous at $a$ and $b$. How can I prove that $$\lim_{h \to 0} \int_{a}^{b} \frac{f(x+h)-f(x)}{h}dx = f(b)-f(a)$$
I was thinking of using the fundamental theorem of calculus, but we don't have the condition that $f$ is continuous, only that $f$ is continuous on $a$ and $b$
Hint: For $|h|$ small and not $0$, your expression is equal to $$f(b)-f(a)+\frac{1}{h}\int_b^{b+h}(f(t)-f(b))dt-\frac{1}{h}\int_a^{a+h}(f(t)-f(a))dt$$