I am reading "Basics of Manifolds" (in Japanese) by Yukio Matsumoto.
The following exercise is Exercise 4.2 on p.25 in this book.
4.2
A function $f$ on an open set $U$ in $\mathbb{R}^m$ is a $C^r$-function ($r\geq 1$) if and only if $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are $C^{r-1}$-functions.
The author's answer to this exercise is here:
An $r$th order partial derivative of $f$ is obtained by partially differentiating one of $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ $(r-1)$ times with respect to certain variables.
Therefore if $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are all $C^{r-1}$-functions, then any of $r$-th order partial derivatives of $f$ is continuous.
So, $f$ is a $C^r$-function.
The converse is obvious.
I think we need to prove $f$ is continuous if $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are all $C^{r-1}$-functions.
If $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are all $C^{r-1}$-functions, then $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are all continuous.
So, it is sufficient to prove $f$ is continuous if $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are all continuous.
And I proved $f$ is continuous if $\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_m}$ are all continuous.
Am I right? (or is this fact obvious?)
