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!

No, what about $f(x)=x$ if $0\le x<1$ and $f(1)=1/2$. Clearly this has the property you stated with $a=0,b=1$ but is not continuous in $[0,1]$.
