Don't you need the set of countable points to be dense in R^n for this to work? Cheers Seb On Dec 21, 2015 12:27 PM, "Gareth McCaughan" <gareth.mccaughan@pobox.com> wrote:
Analytic functions are described by a countable number of coefficients.
It is a delusion to think there is an uncountable number of degrees of freedom.
So to make that clear, what we want is some theorem saying: Specifying the value of the analytic function at a countable set of points within the disk, will suffice to specify the function.
Is such a theorem known? I can't think of one off the top of my head. But here, voila, I proved it:
We can make it much easier than that, while proving a stronger theorem.
Theorem: a (merely) continuous function from R^n to R^n is determined by countably many numbers.
Proof: specify its values at rational points (of which there are only countably many); its values elsewhere are determined by continuity.
-- g
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