This question was asked in my Mid term exam held last week and I was unable to solve it. So, I am looking for some help here.
Show that every bijection $f : \mathbb{R} \to [0,\infty)$ has infinite many points of discontinuity.
I am at loss of ideas. There are some results in the functions of bounded variation like (If f is monotonic on [a,b] , then the set of discontinuities of f is countable) but none of them could be used here.
I thought of letting that $f: \mathbb{R} \to[0 ,\infty$) has only finite many points of disconuity but was unable to think how should I proceed for any contradiction.
Can you outline a argument?