i have a function f:R->R where f is monotone increasing, i need to show that the set of discontinuous points of f is at most countable.(adsbygoogle = window.adsbygoogle || []).push({});

so i need to find an injective or 1-1 mapping from this set to the naturals, or to the rationals.

i thought perhaps defining the next function g:A->Q, where A is the set of discontinuous points of f, by:

let x0 be in A, so lim(x>x0)f(x)>lim(x<x0)f(x)

g(x0)=x0 if x0 is in Q

but how do i define for points which arent in Q?

thanks in advance.

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# Set of discontinuous points.

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