Partial derivative exists with respect to x and doesn't exist with respect to y

Question

I want to find a function whose partial derivative exists with respect x for all $(x,y)\in R^2$ and doesn't exist with respect to y at infinitely many points.

I could only think of functions with absolute value for y, for example: $(,)=^x+||$, but i think that this function doesn't have partial derivative with respect to y only at the origin, (0,0), and not all points as the question suggests! How should i think in this case?

Thanks in advance!

Answer 1

Your problem is not to construct such a function in $\mathbb{R}^2$. It is to find a function that is not differentiable at infinetly many points in one dimension. The hardcore answer are functions that are nowhere differentiable (cf Brownian motion) but these are in general not easy to construct. An easy answer is to think about necessary conditions for a function to be differentiable. Maybe try to construct a function with infinitly many discontinuities.

Answer 2

To get an infinite number of discontinuities, there are lots of things like $f(x,y)=x+\lfloor y\rfloor$ or similar. The partial derivative $\partial f/\partial y$ does not exist at integer values of $y$ but $\partial f/\partial x$ is always $1$. You can also use more zany things like the Dirichlet function in the $y$ direction. Basically anything as long as it doesn't involve $x$ will do.