intermediate value property intuition Let's say we are given a graph. How do we determine from the graph that the function has  intermediate value property? And also, why do we need to impose the condition 'monotone' so that a monotone function with intermediate value property is continuous ? Intuitively, I don see why the statement 'general function has intermediate value property is continous' is false.

The famous example involves the derivative of  the function $f$ defined by $f(x)=x^2\sin(1/x)$ if $x\ne 0$ and $f(0)=0$. 
It is not difficult to show that $f$ is differentiable everywhere, but $f'(x)$ is not continuous at $0$. And, like any derivative, $f'(x)$ has the Intermediate Value Property. 
Remark: The question about intuition is difficult to address. One problem is that continuous functions can be a lot weirder than any "can be drawn without lifting the pencil" function of our imagination. The compromise is an intuitive notion of smoothness, tempered by knowledge of standard counterexamples. 
