I am reading a Multivariable Calculus text written in Japanese. The definition of the functions of Class $C^k$ in the book is here:
If $f(x, y)$ is continuous and has all the partial derivatives of order $1, 2, \cdots, k$ and all the partial derivatives of order $1, 2, \cdots, k$ are continuous, then $f(x, y)$ is called a function of Class $C^k$.
I thought the definition was strange and I checked another multivariable calculus text written in Japanese. And I found the same definition in another text.
Why do the authors adopt the above definition?
I think the definition below is simple and good:
If $f(x, y)$ has all the partial derivatives of order $1, 2, \cdots, k$ and all the partial derivatives of order $k$ are continuous, then $f(x, y)$ is called a function of Class $C^k$.
Observe: if $k \ge 1$ and $f$ has all the partial derivatives of order$1,2,...,k$ and all the partial derivatives of order $1,2,⋯,k$ are continuous, then the partial derivatives of $f$ of order $1$ are continuous.
Then $f$ is (total) differntiable, hence continuous. Hence, the requirement that f is continuous in your japanese book is superfluous