Derivative AT a Discontinuity
If \(f\) is differentiable at \(a\), then it is continuous at \(a\). But what if it’s NOT even continuous? Then how the hell can it be differentiable?
What’s its derivative? Forget the formal definition of the derivative for a moment, and just consider the step function and the intuitive idea of a derivative as a slope.
Look at the step function, left to right. At 0, supposing for a moment there is a derivative, then it can’t be any standard number. It’s not 0 since it’s certainly not flat. It’s not negative since it’s increasing to the right. It’s bigger than 1, 2, 3, 10000000000…
Since it goes from 0 to 1 in the space of a single point, the derivative would have to be infinitely big because its difference quotient is \(\frac{1 - 0}{0 - 0} = \frac{1}{0} = \infty\). The Problem is that no real number is infinitely big.
The usual way mathematicians deal with the Problem is to introduce generalized functions. They are also called distributions, which is confusing since probability distributions are completely different.
