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

##### 1 Answer

See the explanation section below.

#### Explanation:

**Differentiability**

Theorem: If

So, no. If

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.