02 - Differentiability Implies Continuity

Differentiability Implies Continuity

Assume $f:\Omega \subseteq {\mathbb{R}}^n>.{\mathbb{R}}^m$ is differentiable at $\underline{a}\subseteq \Omega$ then

$$f\text{ is continuous at }\underline{a}$$

Proof

Let the derivative be $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ then

$$f(\underline{a}+\underline{h})-f(\underline{a})=\underline{L}h+R(\underline{h})|\underline{h}|$$

where $R(\underline{h})\to \underline{0}$ as $\underline{h}\to \underline{0}$

Taking $\underline{h}\to 0$ on the right hand side gives $\underline{0}$ hence

$$f(\underline{a}+\underline{h})-f(\underline{a})\to 0\text{ as }\underline{h}\to \underline{0}$$