Does continuous second order partial derivatives at $(x_0,y_0)$ in open region $\mathbb{R}^2$ imply continuous first order partial derivatives in same open region $\mathbb{R}^2$?
Continuous second order partial derivates in open region R
4
$\begingroup$
multivariable-calculus
1 Answers
4
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.