I'm working on this problem, I've tried to solve it by using the definition for partial derivative but haven't been able to prove it. Appreciate any help:
Prove that if all partial derivatives of order $m$ of a scalar field $f$ over $\mathbb{R^{n}}$ are continuous in a region $S$, then $f$ and all of its partial derivatives of order less than $m$ are also continuous on $S$.
Thanks!
Just for simplicity consider $f:\mathbb{R}^2\to \mathbb{R}$ and assume that $\frac{\partial^2 f}{\partial x^2},$ $\frac{\partial^2 f}{\partial x \partial y},$ $\frac{\partial^2 f}{\partial y \partial x},$ and $\frac{\partial^2 f}{\partial y^2}$ are continuous on some domain. Now, by definition, it is $$\frac{\partial^2 f}{\partial x^2}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right), \frac{\partial^2 f}{\partial y\partial x}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right).$$ This implies that $\frac{\partial f}{\partial x}$ exists and it is derivable with respect to $x$ and $y,$ being the partial derivatives continuous on the given domain by assumption. So $\frac{\partial f}{\partial x}$ is differentiable and, as consequence, continuous. In a similar way we show that $\frac{\partial f}{\partial y}$ is differentiable and, as consequence, continuous. Finally, using again the differentiability theorem, we get that $f$ is differentiable and, consequently, continuous.
I hope you can generalize the argument to higher dimensions and higher order partial derivatives.