Main Question or Discussion Point
What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?
I'm not sure about this. Isn't [itex]F(x)= \int_0^x |t|dt[/itex] differentiable at 0? It is the piecewise function given by F(x)=x^2 for x>0 and F(x)=-x^2 for x>0 and F(0)=0.Those two examples also have the property that while [itex]F(x)= \int f(t)dt[/itex] is defined, F(x) itself has no dervative at x= 0.