Q:

Continuous second order partial derivates in open region $R$

Does having continuous second order partial derivatives at a point $(x_0,y_0)$ in an open region of $\mathbb{R}^2$ imply having continuous first order partial derivatives in the same open region $\mathbb{R}^2$? 

A:

There is a theorem that states "if a function is differentiable at a point, then it is continuous at that point." Consider the first order partial derivative is just some function with its own graph. So, if its derivative (second-order partial derivative) is continuous at some point $(x_0,y_0)$ then the function itself (first-order partial) is continuous at that point.

