Differentiability ⇒ continuity

Differentiability implies continuity

Let $X$ be a subset of $R$ , and let $f : X \to R$ 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 $R$ and let $f : X \to R$ 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'.

14.04.2020