Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.
Let $h : \mathbb{R} \to \mathbb{R}$ be defined so that $h|_{[0, 1]} : x \mapsto x$, $h|_{[1, 2]} : x \mapsto 2-x$, and for all $x \in \mathbb{R}$, $h(x) = h(x+2)$. Note that $h$ behaves as a triangle wave, and that $h(x) = h(-x)$.
Define a function $f$ as follows: $$\begin{align*}f : \mathbb{R} &\longrightarrow \mathbb{R} \\ t &\longmapsto \begin{cases} h\left(\frac1x\right) & \text{ if } x \neq 0 \\ 0 & \text{ if } x = 0. \end{cases}\end{align*}$$
What we've shown here is that not every function which satisfies the conclusion of the intermediate value theorem is continuous. In fact, things can be much, much worse: the Conway base 13 function takes on every real value in every non-empty open interval — in particular, it has the intermediate value property — but is horribly discontinuous at every point.