# Intermediate Value Functions

A continuous function $$f\colon\mathbb{R}\to\mathbb{R}$$ 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 $$f\restriction [a,b]$$ takes all possible values between f(a) and f(b)). Without words:
$$\forall a,b\in\mathbb{R}$$
$$\quad\forall x(f(a)\leqslant x \leqslant f(b)\lor f(b) \leqslant x\leqslant f(a))\rightarrow (\exists y\in [a,b] f(y)=x)$$

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

Is an iv-function necessarily continuous?

2/5