TITLE: Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?
QUESTION [48 upvotes]: Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?  
It seems to me like they are equal definitions in a way.  
Can you give me a counter-example?  
Thanks

REPLY [12 votes]: The next theorem might be of interest to you, it really shows that the class of functions with the IVP is very big.
Theorem (Sierpinski) Let $f : \mathbb R \to \mathbb R$ be any function. Then there exists $f_1,f_2 : \mathbb R \to \mathbb R$ such that $f=f_1+f_2$ and $f_1,f_2$ satisfy the Intermediate Value Property.
Moreover, in the above Theorem, $f_1,f_2$ can be chosen to be discontinuous at all points.