How can I prove the continuity of an injective function that satisfies Intermediate Value property by sequential criteria

Question

I am trying to prove the continuity of a function which is injective and satisfies the Intermediate Value property.I want to proceed by using sequential criteria and by using the method of contradiction.

Answer 1

We prove a bit more general result: Let $f:(\alpha ,\beta )\to\mathbb{R} $ be a function such that for every $w\in\mathbb{Q}$ the set $f^{-1} (\{w\})$ is closed. Then the function $f$ is continuous.

${\it Proof.}$ Suppose that $f$ is not continuous, then there exists a point $x_0$ of discontiniuity of the function $f.$ Hence there exists a sequence $x_n\to x_0$ and a number $g\in\overline{\mathbb{R}}$ (the extended real line) such that $f(x_n )\to g.$ Without loss o generality we may assume that $f(x_0 )<g$ and let $q$ be a rational from interval $(f(x_0), g).$ By the Intermediate Value property we may construct a sequence $v_n\to x_0$ such that $f(v_n )=q$ hence the set $f^{-1} (\{q\})$ is not closed.