General form of functions

Question

Is there a general form for functions? For example if the function is a polynomial, the general form is well-known. But is there a general one, covering every possible function?

Answer 1

I suppose it depends on what you mean by a 'general form', but regardless of your definition, the answer will almost certainly be no. Let's assume you're talking about functions $\mathbb{R} \to \mathbb{R}$.

Consider polynomials, these have a general form which is specified by finitely many real numbers (the coefficients of $x^n$). It is a nice exercise to show that the set of finite sequences of real numbers has a strictly smaller cardinality that the set of all functions $\mathbb{R} \to \mathbb{R}$, denoted $\mathbb{R}^{\mathbb{R}}$. Likewise, many other named families of functions are specified by finitely many real numbers, and therefore suffer the same fate.

What about something like power series? Here you can take an infinite sum of the functions $x^n$, $n \geq 0$, together with real number coefficients. That is, you specify a real number for every $n \in \mathbb{N}$. Regardless, you are still far from capturing all the functions. More precisely, the cardinality of the set of all infinite sequences of $\mathbb{R}$, denoted $\mathbb{R}^{\mathbb{N}}$, has cardinality strictly smaller than $\mathbb{R}^{\mathbb{R}}$.

So we've seen that you can't describe every element of $\mathbb{R}^{\mathbb{R}}$ using finitely many real numbers, or indeed by specifying a real number for every natural number. So how many real numbers do you need to specify to be able to describe every element of $\mathbb{R}^{\mathbb{R}}$? Well, by playing around with cardinalities (which is what all of the above is really about) we find that we need to specify a real number for every real number. That is, for every $x \in \mathbb{R}$, you need to specify $f \in \mathbb{R}$. This notation is a bit misleading as the real number $f$ that you have to choose could change depending on $x$, so instead write $f(x)$. Now you see that you have specified an element of $\mathbb{R}^{\mathbb{R}}$. If you make some appropriate choices, you can realise this specification as simply describing all of the values of any given function. I don't know about you, but that doesn't sound all that general to me.

Answer 2

To give a bit more detail about Michael Hardy's point:

Suppose a "description" or "form" of a function consists of a finite string of symbols chosen from a finite alphabet. Then it turns out that the set of all descriptions is countable (essentially, each can be seen as representing a natural number in some base).

Let $S$ be a set with at least two elements, and let $T$ be an infinite set. Then $S^T$ will have a cardinality of at least $|2^{\aleph_0}|$, the cardinality of the set of infinite bit strings. Cantor's diagonal argument shows this to be uncountable. So there isn't even any way to describe all functions from the set $\Bbb N$ to the set $\{0,1\}$, let alone something more interesting like the functions from the reals to the reals!