Differentiability from continuous partial derivatives ($Df(a)h={\sum }_j{\partial }_{j}f(a)h_j$)

If all partial derivatives exist near a point and are continuous there, successive coordinate increments and the one-variable mean value theorem express $f(a+h)-f(a)$ as ${\sum }_j{h}_j{\partial }_{j}f(a+{\eta }_j(h))$ with ${\eta }_j(h)\to 0$. Subtracting the displayed linear map leaves $o(\Vert h\Vert )$, proving Fréchet differentiability. In two dimensions the absolute remainder is at most $ϵ(|h_1|+|h_2|)\le \sqrt{2}ϵ\Vert h\Vert$ for small enough increments.