Math and science::Analysis::Tao::10: Differentiation of functions

# Differentiability at a point, definition

### Differentiability at a point

Let $$X$$ be a subset of $$\mathbb{R}$$, and let $$x_0 \in X$$ be an element of $$X$$ that is also a limit point of $$X$$. Let $$f : X \to \mathbb{R}$$ be a function. Then, if the following limit:

[...]

converges to some $$L$$ then we say that $$f$$ is differentiable at $$x_0$$ on $$X$$ with derivative $$L$$. We write: $$f'(x_0) := L$$.

If $$x_0$$ is not an element of $$X$$ or is not a limit point of $$X$$, or the limit does not converge, we leave $$f'(x_0)$$ undefined and we say that $$f$$ is not differentiable at $$x_0$$ on $$X$$.

We need $$x_0$$ to be a limit point (not isolated) in order for it to be adherent to $$X \setminus \{x_0\}$$ and in turn for the above limit to be defined. Thus, functions do not have a derivative defined at isolated points.