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Math Equations

Sunday, February 1, 2015

How many functions are there?

Claim
There are more real functions than there are real numbers.

Explanation
{f:RR}{f:R{0,1}}=P(R)>R

Claim
There are just as many continuous functions as there are real numbers.

Explanation
{f:RR,f continuous}{{f:{0}R}=R{f:N×NR}=RNN=2NNN=2N=R

The first step on the second line is justified by the fact that for every continuous function, there is a sequence of parallelogram functions converging pointwise to it. By parallelogram function, I mean a function whose value is specified at evenly distributed points and interpolated linearly in between. For example, a parallelogram function approximating sin(x) is f(x)=sin(x)+(xx)sin(x+1).

To narrow the number of parallelogram functions down to {f:N×NR}, require that the value at 0 be specified and that the distance between specified points for the nth function in the sequence be 1/n.

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