$f(x)$ is everywhere differentiable on $[a,b]$ then give examples for each (they are independent)
(1) $f'(x)$ is not Riemann integrable
(2) $f''(x)$ does not exist
(3) $f'(x)$ is not continuous
$f(x)$ is everywhere differentiable on $[a,b]$ then give examples for each (they are independent)
(1) $f'(x)$ is not Riemann integrable
(2) $f''(x)$ does not exist
(3) $f'(x)$ is not continuous
(1) is a famous pathological function called Volterra's function, you can read about it on Wiki at: http://en.wikipedia.org/wiki/Volterra%27s_function It's a pretty interesting function IMO
(2) seems like the second integral of something like $\sin(1/x)$ would work http://www.wolframalpha.com/input/?i=double+integral+of+sin%281%2Fx%29+ so that $f''(x) = \sin(1/x)$ and it isn't defined at $x = 0$;
(3) Edit: as per Ian's suggestion, the Volterra function works for this as well and abs(x) doesn't