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 an element of X and a limit point of \( X \). 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'.

Source

p255