Question

Just because a function exists at a given value does not mean that it is necessarily differentiable at that value. Can a function be differentiable at a discontinuity?

Answer 1

You cannot differentiate at a discontinuity, because a point must exist, and for a point to exist, it must not be a discontinuity.

Another way to think about it.
The derivative is the rate of change. Can you take the rate of change when there is no point?

Answer 2

Thank you for your answer. Exactly what I was thinking.