multivariable calculus - What does it mean for partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ then $f$ is differentiable at $(a,b)$?

I am reading my text book and I come across a theorem that says:

If the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ then $f$ is differentiable at $(a,b)$.

What does it mean for partial derivatives $f_x$ and $f_y$ exist near $(a,b)$? If I just find the partial derivatives of $f_x$ and $f_y$, does that mean they exist near $(a,b)$? How do I check?

And how do I check to see if they are continuous at $(a,b)$? Can I just plug in a given point, and if get a finite answer, that means it's continuous at $(a,b)$ correct? But then there might be a gap.... so how do I know for sure with a given point? When would it not be continuous, when infinity, etc. ?

Thank you