Differentiability of vector valued function? The question is:
Let $g$ and $h$ be real-valued functions on $R$ such that $g$ is differentiable at "a" and $h$ is differentiable at "b". Show that $f$: $R$2 → $R$ defined by $f$( x1, x2) = $g$(x1)$h$(x2) is differentiable at $x$o = (a,b).
Since $g$ and $h$ seem to be described as "univariate", my idea for a solution was to basically prove the product rule for two functions. Then use that result to justify the differentiability of $f$... I hope


Lemma: Given $\phi:\mathbb{R}^2 \to \mathbb{R}$ and $\psi: \mathbb{R}^2 \to \mathbb{R}$ differentiable, then $\phi \cdot \psi$ is differentiable.
Proof: $\phi \cdot \psi=m \circ (\phi \times \psi),$ where $m(x,y)=x \cdot y$. Therefore, by the chain rule, $\phi \cdot \psi$ is differentiable.

$f=(g \circ \pi_1) \cdot (h \circ \pi_2)$. Since projections are linear, they are differentiable. Since composition of differentiable functions is differentiable (by the chain rule), then the result follows from the lemma.
