I have two questions that i'm curious about.
If $f$ is differentiable real function on its domain, then $f'$ is Riemann integrable.
If $g$ is a real function with intermediate value property, then $g$ is Riemann integrable.
Thank you in advance.
I have two questions that i'm curious about.
If $f$ is differentiable real function on its domain, then $f'$ is Riemann integrable.
If $g$ is a real function with intermediate value property, then $g$ is Riemann integrable.
Thank you in advance.
Hints:
1) The function $f$ defined by $f(x)=\cases{ x^2\sin(1/x^2),&$x\ne0$ \cr 0,&$x=0$}$ is differentiable on $[-1,1]$; but its derivative is unbounded on $[-1,1]$.
2) Derivatives enjoy the Intermediate Value Property (by Darboux's Theorem).