If we have a single-variable function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ it is quite clear what it means when we say $f$ is $C^k$ (or of class $C^k$), $k \in \mathbb{Z}, k \ge 0$:

- when $k \ge 1$ that means all the derivatives of $f$ up to the $k$-th one exist, and in addition to that $f^{(k)}$ is continuous
- when $k=0$ that means simply that $f$ is continuous

Now suppose we have the multi-variable function $$f : \mathbb{R}^m \rightarrow \mathbb{R}$$

Here $m \in \mathbb{N}$ and $m \ge 2$.

In that case what does it mean when we say that $f$ is of class $C^k$, $k \in \mathbb{Z}, k \ge 0$?

I met this concept on page 145 of the Vector Calculus book by Baxandall and Liebeck, and I realized that this concept is not quite clear to me. So I did some research on that, and it turned out this concept pertains to the partial derivatives of $f$.

The best definition that I found is the one given here.

So in the multi-variable case, it turns out that

- when we say $f$ is of class $C^k$, $k \in \mathbb{N}$, that means that all the partial derivatives of $f$ of order $s \le k$ exist and are continuous
- when we say $f$ is of class $C$ or $C^0$, that means just that $f$ is continuous

Finally, when the function $f$ is of class $C^1$, it is also called continuously differentiable.

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