1
$\begingroup$

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 $\mathbb R^2$ ?

$\endgroup$
  • 6
    $\begingroup$ 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. $\endgroup$ – Pietro Majer Nov 4 '13 at 14:12
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy