Let $f : \mathbb{R} \rightarrow \mathbb{R} $ with intermediate value property and increasing over $ \mathbb{R} $ \ $\mathbb{Q}$.

Question

Let $f : \mathbb{R} \rightarrow \mathbb{R} $ with intermediate value property and increasing over $ \mathbb{R} $ \ $\mathbb{Q}$. Then $f$ is continous on $\mathbb{R}$. How to try?

Hint: first, show that for $q\in \Bbb Q$, $\lim_{x\to q, x < q, x\in \Bbb R\setminus \Bbb Q}f(x)$ exists. — Didier, Mar 29 '22 at 14:14
@Didier My problem is how to use that the functions has intermediate value property. — Almath, Mar 29 '22 at 14:22
You should then edit your question and post what you have done so far (with details) so that we can give you some advices accurately — Didier, Mar 29 '22 at 14:23

score 0 · Answer 1 · answered Mar 29 '22 at 15:05

Let $x$ be real. Take an irrational number $\alpha$ such that $\alpha >x$. Clealry $f(\alpha)\ge f(x)$, for if not, then $$(f(\alpha),f(x))\subseteq f[(x,\alpha)]\cap \mathbb{Q}\subseteq \mathbb{Q}$$ by our hypotesis, which is absurd. By the same argument $f(\alpha)\le f(x)$ if $\alpha$ is irrational and $\alpha <x$. Hence $f$ is increasing on the whole line. Can you take it from here?

Let $f : \mathbb{R} \rightarrow \mathbb{R} $ with intermediate value property and increasing over $ \mathbb{R} $ \ $\mathbb{Q}$.

1 Answers1