Does there exist a function that is differentiable but not integrable? or integrable but not differentiable?

It has become very complicated to me to find out a function which is differentiable but not integrable or integrable but not differentiable.

Solution 1:

A low level answer:
Differentiable Function that is not Integrable: $f(x)=\frac{1}{x}$ is differentiable in $(0,1)$, however it's not integrable in this interval. $$\int_{0}^{1} f(x) \mathrm dx=\int_{0}^{1} \frac{1}{x} \mathrm dx$$ $$=\ln 1- \ln 0$$ That's Undefined.
Integrable but Not-Differentiable Weierstrass Function (Repeating what already answered).

Solution 2:

Well, If you are thinking Riemann integrable, Then every differentiable function is continuous and then integrable!

However any bounded function with discontinuity in a single point is integrable but of course it is not differentiable!