Find a function whose first partial with respect to $x$ does not exist.

Question

I'm having a difficult time understanding what it means for a first partial derivative of a function to exist. I was wondering if someone could come up with an example of a function defined in the $xy$-plane whose first partial with respect to $x$ does not exist, and show this using the definition of derivative?

Answer 1

Hint: Find a function $g: \mathbb R \to \mathbb R$ whose derivative doesn't exist somewhere, and then define $f(x,y)=g(x).$

Answer 2

How about $f(x,y)=\ln x$? The partial derivative $\frac{\partial}{\partial x}f(x,y)=\frac1x$ at $(0,y)$ doesn't exist.