There are more real functions than there are real numbers.
Explanation
{f:R→R}≥{f:R→{0,1}}=P(R)>R
Claim
There are just as many continuous functions as there are real numbers.
Explanation
{f:R→R,f continuous}{≥{f:{0}→R}=R≤{f:N×N→R}=RN⋅N=2N⋅N⋅N=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⌋)+(x−⌊x⌋)sin(⌊x⌋+1).
To narrow the number of parallelogram functions down to {f:N×N→R}, 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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