= Solution
A function is a <differentiable function> at $a_0=(a,b)$ if there is a <linear map> $L:\mathbb R^2\to\mathbb R$ with
$$
f(a_0+h)=f(a_0)+Lh+r(h),\qquad\frac{r(h)}{\|h\|}\longrightarrow0\quad(h\to0).
$$
The map $L=Df_{a_0}$ is its derivative; equivalently $L(h_1,h_2)=Ah_1+Bh_2$ for fixed constants. Since every such linear map is bounded, $|f(a_0+h)-f(a_0)|\leq\|L\|\|h\|+|r(h)|\to0$. This proves \b[<differentiability implies continuity>], not merely continuity along each coordinate axis.
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