Paddy Leahy writes:
> Oops, mea culpa. I said that wrong. What I meant was, what is the
> cardinality of the data needed to specify *one* continuous function of the
> continuum. E.g. for constant functions it is blatantly aleph-null.
> Similarly for any function expressible as a finite-length formula in which
> some terms stand for reals.
I think it's somewhat nonstandard to ask for the "cardinality of the
data" needed to specify an object. Usually we ask for the cardinality
of some set of objects.
The cardinality of the reals is c. But the cardinality of the data
needed to specify a particular real is no more than aleph-null (and
possibly quite a bit less!).
In the same way, the cardinality of the set of continuous functions
is c. But the cardinality of the data to specify a particular
continuous function is no more than aleph null. At least for infinitely
differentiable ones, you can do as Russell suggests and represent it as
a Taylor series, which is a countable set of real numbers and can be
expressed via a countable number of bits. I'm not sure how to extend
this result to continuous but non-differentiable functions but I'm pretty
sure the same thing applies.