Converse Of Intermediate Value Theorem

Calculus Level 2

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

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

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

The intermediate value theorem states that if ff is continuous, then ff 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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