**Please comment your solutions, questions and remarks.**.

A continuous function [tex]f\colon\mathbb{R}\to\mathbb{R}[/tex] on the real numbers has the following **property**: Given any two reals a<b, the function f gets all possible values between f(a) and f(b) on the interval [a,b] (i.e. the function [tex]f\restriction [a,b][/tex] takes all possible values between f(a) and f(b)). Without words:

[tex]\forall a,b\in\mathbb{R}[/tex]

[tex]\quad\forall x(f(a)\leqslant x \leqslant f(b)\lor f(b) \leqslant x\leqslant f(a))\rightarrow (\exists y\in [a,b] f(y)=x)[/tex]

Let us call functions with the described **property** *iv-functions* (for intermediate value).

Is an iv-function necessarily continuous?

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