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

Answer 1

See the explanation section below.

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.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

If a function has a removable discontinuity, it is still differentiable at that point. However, it may not be integrable at that point.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7