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

Differentiability ⇒ continuity

Differentiability implies continuity

Let $$X$$ be a subset of $$\mathbb{R}$$, and let $$f : X \to \mathbb{R}$$ be a function. Let $$x_0$$ be [...]. If $$f$$ is differentiable at $$x_0$$ on $$X$$, then $$f$$ is continuous at $$x_0$$.

Tao's definition of function continuity includes the assumption that the limit is being taken over $$X$$, hence why 'on $$X$$ is missing from the end of the proposition above.

The definition of differentiation on a domain along with the proposition above that differentiability implies continuity brings us to an immediate corollary.

Let $$X$$ be a subset of $$\mathbb{R}$$ and let $$f : X \to \mathbb{R}$$ be a function. If $$f$$ is differentiable on $$X$$, then $$f$$ is continuous.

I'm not quite sure why Tao decides to now include the idea of 'continuity on X'.