Can $f_x$ be undefined while $f_y$ is defined?

Question

While studying the critical points of two variable functions, I noticed that my textbook mentioned that if one partial derivative is undefined at a point, then that point is a critical point. I was wondering if it possible for one partial derivative to exist and another to not exist?

Answer 1

Consider the partial derivatives of the function $$f(x,y)=|x|$$ at the point $(0,0)$.

Answer 2

Sure, let $f(x,y) = \sqrt[3]{x} y$, then at $(0,0)$ $f_x$ is not defined while $f_y$ is.