are the partial derivatives must be continuous in the chain rule? Let $g(t)=(x(t),y(t))$ and suppose that $g'(t)$ exist in $t=t_0$. By the chain rule, if the partial derivatives of a function $f(x,y)$ are continuous in an open neighborhood of $g(t_0)=P$, then 
$$
(f\circ g)'(t_0)=f_x(P)x'(t_0)+f_y(P)y'(t_0).
$$
Are the partial derivatives $f_x$ and $f_y$ must be continuous in $P$ or there exist a version of the chain rule with weaker conditions? 

Let $A\subset\mathbb{R}^{n},\ B\subset\mathbb{R}^{m}$ open sets and let $f:A\to\mathbb{R}^{m},\ g:B\to\mathbb{R^{l}}$ be functions such that:


*

*$f$ is differentiable at $a\in A$

*$g$ is differentiable at $f(a)$

*$f(A)\subseteq B$


Then, $g_{o}f$ is differentiable at $a$ and $(g_{o}f)'(a)=g'(f(a))f'(a)$.
