calculus - Can function be differentiable but not continuous?

Is there any possible function that is not continuous but differentiable?

For example these functions, $f(x) = \pi x + \pi $ whenever $x<0$ ,and $f(x) = \arctan \pi x$ when $0\leq x$.

I know that these are not continuous but when I derivative them, they get the same answers. Should I consider that differentiable or not?