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

See the explanation section below.

Differentiability

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.

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

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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.

- What is the limit of #(4x − sin4x)/(4x − tan4x)# as x approaches 0?
- I get a solution of #y=3# but how to solve for this limit?? #lim(x->-4)(3x+12)/(x+4)#
- How do you evaluate the limit #lim((x-3)^2-9)/(2x)dx# as #x->0#?
- How do you find the horizontal asymptote of the graph of #y=6x^2# ?
- Given #x^2 + 2# how do you find the limit as x approaches 3?

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