A friend recently supposed that there were cc continuous functions. (c is the number of reals.) I replied that there were only c continuous functions, but could not immediately remember why. Here is why:
If you know the value of f(x) for every rational x, then by continuity you know f(x) for all x. There are only a countable number of rationals. Let rn be the nth rational in an enumeration that exhausts the rationals. Let Fn = f(rn). There are altogether a denumerable number of digits in the decimal expression of all the F’s collectively. There are only c such denumerable sequences of digits. Thus there are only c such F sequences and thus c continuous functions.
Similarly there are only c monotonic functions. There are only a denumerable number of discontinuities in a monotonic function for each defines a range of values and there can be only a denumerable number of disjoint ranges on the real line. If we know the values of a monotonic function at each rational and at each discontinuity then we have the same argument as for continuous functions above.
The points in Hilbert space are more and less than the continuous functions. There are two common isomorphic incarnations of Hilbert space. A point of one incarnation is merely a denumerable sequence of reals (as in Fourier coefficients) and there are thus c such points as in the argument above. The other incarnation is composed of square integrable functions which need not be continuous. There are more than c discontinuous functions but there is an equivalence relation that confounds two functions when the integral of the square of their difference is zero. Points of Hilbert space are actually these equivalence classes of which there are only c.
For a topological space X, such as the domain of our functions, a base is a set B of sub-sets of X such that for any point x in X and any positive real z, there is an element b of B such that every point in b is no farther than z from x.
If I take a denumerable base over the domain of the functions, and compute the integral of the square of the function over each of these base elements, I again have a denumerable set of reals that determine the function, even if it is discontinuous. Such a denumerable base is every ball of radius 2−n (for all n) about each point with rational coordinates.
To turn this into a proof requires a half dozen theorems each with slightly non trivial proofs. Such is the content of the first few weeks of a course on Hilbert space.