Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere? 
Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

I think it is probable because we can consider 
$$  y = 
\begin{cases}
\sin \left( \frac{1}{x} \right), & \text{if } x \neq 0, \\
0, & \text{if } x=0.
\end{cases}
$$
This function has intermediate value property but is discontinuous on $x=0$.
Inspired by this example, let $r_n$ denote the rational number,and define 
$$  y = 
\begin{cases}
\sum_{n=1}^{\infty} \frac{1}{2^n} \left| \sin \left( \frac{1}{x-r_n} \right) \right|, & \text{if } x \notin \mathbb{Q}, \\
0, & \mbox{if }x \in \mathbb{Q}.
\end{cases}
$$
It is easy to see this function is discontinuons if $x$ is not a rational number. But I can't verify its intermediate value property.

Not only such functions exists, but it turns out there are lots of such functions. 
There is actually a Theorem, I think by Sierpinski, that any real valued function is the sum of two Darboux functions...
