calculus - Does Intermediate Value Theorem $rightarrow$ continuous?

i try to understand Intermediate Value Theorem and wonder if the theorem works for the opposite side. I mean, if we know that $\forall c\:\:\:f\left(a\right)\le \:c\le \:f\left(b\right)\:,\:\exists x_0\in \left[a,b\right]\:\:$ such that $f\left(x_0\right)=c$ then $f\:$ is continuous in $\left[a,b\right]$? tnx!

EDITED: Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? there is absolute fantastic answer for this!