Continuous Differentiability

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.

Partial Derivatives Notation

Suppose that we have the function $$f : \mathbb{R}^2 \rightarrow \mathbb{R}$$ which is a function of two real variables. 

We denote its second order partial derivatives in the following way

$$\frac{\partial^2 f}{\partial x^2}$$

$$\frac{\partial^2 f}{\partial y^2}$$

$$\frac{\partial^2 f}{\partial x \partial y}$$

$$\frac{\partial^2 f}{\partial y \partial x}$$

In this notation it is important to remember that by definition

$$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial f}{\partial x}\left(\frac{\partial f}{\partial y}\right)$$

$$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial f}{\partial y}\left(\frac{\partial f}{\partial x}\right)$$

So if we write x first in the denominator, that means x is the second variable we differentiate by, 

and if we write y first in the denominator, that means y is the second variable we differentiate by. 

In other words, first we differentiate by the right-most variable (from the denominator), second we differentiate by the second right-most variable (from the denominator) and so on. Of course these are not really denominators (in the algebraic sense) but OK, they look like denominators.

The same applies to functions of more than two variables.

Suppose $$g : \mathbb{R}^3 \rightarrow \mathbb{R}$$ is a function of three real variables. 

Then by definition the expression 

$$\frac{\partial^3 g}{\partial x \partial y \partial z}$$ means the following

$$\frac{\partial^3 g}{\partial x \partial y \partial z} = \frac{\partial g}{\partial x}\left(\frac{\partial g}{\partial y} \left(\frac{\partial g}{\partial z}\right)\right)$$

In other words here we have differentiated first by $z$, then by $y$, and finally by $x$.