partially differentiable, but not continuous

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.

• Do you need the function to be non-differentiable (discontinuous) everywhere, or just on some set? Would a single point be enough? Oct 12, 2017 at 15:26
• an open set would be good
– dba
Oct 12, 2017 at 15:49
• 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)? Oct 12, 2017 at 19:25
• every point must have both partial derivatives
– dba
Oct 12, 2017 at 19:28
• 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/… Oct 12, 2017 at 19:49