Intermediate Value Functions

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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