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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Set of discontinuous points.

**Physics Forums | Science Articles, Homework Help, Discussion**