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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 $C^k$.

Teiji Takagi's and Mitsuo Sugiura's Calculus books

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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

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    Thank you very much, Mr. Fred. – 2017-02-10
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    It is a mystery for me that all the Japanese authors adopt the above superfluous definition in their calculus book. A Fields medalist, Kunihiko Kodaira also adopted the above superfluous definition. It's a mystery. – 2017-02-10
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That is done such that the class $C^0$ is not the class of all functions such that a vacuous condition holds, resulting in the class of all functions. Instead, with the definition written as in those books $C^0$ is the class of continuous functions.

If they don't define explicitly $C^0$, it is likely that they inherited the definition from a text that does include $C^0$.