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 $(a,b)$?

Question

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

Answer 1

It means that if there is an open disk $D$ containing $(a,b)$ such that $f_x(x,y)$ and $f_y(x,y)$ exist at every point in $D$, and are both continuous at the one point$(a,b)$, then the function $f(x,y)$ is differentiable at $(a,b)$. Continuous here means that $\lim_{(x,y) \rightarrow (a,b)} f(x,y) = f(a,b)$.

The conclusion is stronger than just that $\partial_xf$ and $\partial_y f$ both exist at $(a,b)$; see the definition of differentiable.

Answer 2

The partial derivatives exists if, when you calculate the partial derivatives of $f$ to get functions $f_x$ and $f_y$, these functions are defined and give you a number at $p$. To check continuity of any function at any point, it will probably be best to the "sequential definition" of continuity.

As a counterexample to the idea you just have to check a function is bounded at a point to get it is continuous, consider the function $f(x) = 1$ if $x$ is rational and zero otherwise.