I try to find a function that is partially differentiable in all points, but not differentiable or even continuous.

I constructed a function where this holds for an uncountable set of Points $$f: \mathbb{R^2} \rightarrow \mathbb{R}, (x,y) \rightarrow \begin{cases} 1, & x\text{ and }y\text{ are rational} \\ 0, & \text{else} \end{cases} $$

If I choose $(x,y)\in \mathbb{R}\backslash \mathbb{Q}$ the partial derivatives exist in these points. But that is not what I am looking for.

  • $\begingroup$ Do you need the function to be non-differentiable (discontinuous) everywhere, or just on some set? Would a single point be enough? $\endgroup$ Oct 12, 2017 at 15:26
  • $\begingroup$ an open set would be good $\endgroup$
    – dba
    Oct 12, 2017 at 15:49
  • $\begingroup$ Do you want every point to have both partial derivatives defined? Would it be acceptable to have at least one partial derivative at every point (but not both at many points)? $\endgroup$ Oct 12, 2017 at 19:25
  • $\begingroup$ every point must have both partial derivatives $\endgroup$
    – dba
    Oct 12, 2017 at 19:28
  • $\begingroup$ Any region in which the partial derivatives are continuous will also have continuity. To get away from that, the partials need to be discontinuous a lot, perhaps on a dense set. This question seems relevant: math.stackexchange.com/questions/292275/… $\endgroup$ Oct 12, 2017 at 19:49


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