How to define functions that have functions as parameters? On the most basic level I would like to define something like this:
$$g:\Bbb R\to\Bbb R, x \mapsto x$$
$$f:\Bbb R\times?\to\Bbb R, (x, g)\mapsto xg(x)$$
My question is: How would one define a function such that another function can be a parameter?
 A: $f: \mathbb R \times {\mathbb R}^{\mathbb R} \to \mathbb R$.
$X^Y$ is the set-theoretic notation for the set of functions from $Y$ to $X$ (mind the order!)
For example $2^{\mathbb N} = \{0,1\}^{\mathbb N} = $ the set of sequences of zeroes and ones.
A: You don't need any special notation for this, the only thing you would like to do is specify the domain. The usual mathematical notation for set of functions $A \to B$ is $B^A$, so you could write like this:
$$f : B^A \to C,$$
but 
$$ f : (A \to B) \to C$$
should be understood as well (especially within the area of computer-science-related math). The definition itself is not special, e.g.
$$f(g) = g(5).$$
However, be aware, that there is a notation for functions returning functions (i.e. having functions as values). As before we could write $f : A \to C^B$ or $f : A \to (B \to C)$, but there are two ways to define it:
$$f(x) = y \mapsto x+y$$
or
$$f(x)(y) = x+y.$$
Finally, you can combine the first part and the second part, for example $f : \mathbb{R}^\mathbb{R} \to \mathbb{R}^\mathbb{R}$ or $f : (\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R})$ and
$$ f(g) = x \mapsto x\cdot g(x) $$
or
$$ f(g)(x) = x \cdot g(x).$$
I hope this helps $\ddot\smile$
