Is such a function differentiable?

Question

Let $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function whose both partial derivatives of the first order exist on a dense vsubset $D\subset \mathbb R^2$ and these partial derivatives extend to continuous functions $f_1,f_2: \mathbb R^2 \rightarrow \mathbb R$. Is it $f$ differentiable or of class $C^1$ on $\mathbb R^2$ ?

Answer 1

Making the comment an answer: no: think to the Cantor function $f(x)$ (as a function of two variables). It has $\partial_x f = 0$ on a dense set, and $\partial_y f=0$ everywhere.