= Differentiability from continuous partial derivatives
{title2=$Df(a)h=\sum_j\partial_jf(a)h_j$}
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 $\sum_j h_j\partial_jf(a+\eta_j(h))$ with $\eta_j(h)\to0$. Subtracting the displayed <linear map> leaves $o(\|h\|)$, proving <Fréchet differentiability>. In two dimensions the absolute remainder is at most $\epsilon(|h_1|+|h_2|)\le\sqrt2\epsilon\|h\|$ for small enough increments.
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