Question

If a function has a removable discontinuity, is it still differentiable at that point? What about integrable?

Answer 1

See the explanation section below.

Explanation:

Differentiability

Theorem: If `f` is differentiable at `a`, then `f` is continuous at `a`.

So, no. If `f` has any discontinuity at `a` then `f` is not differentiable at `a`.

For proof, see any introductory calculus textbook for sciences.
(Not all applied calculus books include the proof.)

Integrability

It depends on the definition of integral at a particular point in a student's education. Some treatments start with the integral of a continuous function on a closed interval. So continuity is a prerequisite for integrability.

Eventually, we do define definite integral in such a way that a function with a removabla discontinuity is integrable.

And a function with a (finite) jump discontinuity is integrable.

And even some functions with infinite discontinuities are integrable.