Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 3 / 3B / Solution

A function is a differentiable function at $a_0=(a,b)$ if there is a linear map $L:{\mathbb{R}}^2\to \mathbb{R}$ withThe map $L=D{f}_{a_0}$ is its derivative; equivalently $L(h_1,h_2)=A{h}_1+B{h}_2$ for fixed constants. Since every such linear map is bounded, $|f(a_0+h)-f(a_0)|\le \Vert L\Vert \Vert h\Vert +|r(h)|\to 0$. This proves differentiability implies continuity, not merely continuity along each coordinate axis.