Wronsonia8g

2022-06-30

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

pampatsha

Expert

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

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