# Converse Of Intermediate Value Theorem

Calculus Level 2

A real-valued function $$f$$ is said to have the intermediate value property if for every $$[a,b]$$ in the domain of $$f$$, and for every

$x \in \Big[\min\big(f(a), f(b)\big), \max\big(f(a), f(b)\big)\Big],$

there exists some $$c\in [a,b]$$ such that $$f(c) = x$$.

The intermediate value theorem states that if $$f$$ is continuous, then $$f$$ has the intermediate value property. Is the converse of this theorem true? That is, if a function has the intermediate value property, must it be continuous on its domain?

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