(a) State and prove the intermediate value theorem.

(b) An interval is a subset $I$ of $\mathbb{R}$ with the property that if $x$ and $y$ belong to $I$ and $x<z<y$ then $z$ also belongs to $I$. Prove that if $I$ is an interval and $f$ is a continuous function from $I$ to $\mathbb{R}$ then $f(I)$ is an interval.

(c) For each of the following three pairs $(I, J)$ of intervals, either exhibit a continuous function $f$ from $I$ to $\mathbb{R}$ such that $f(I)=J$ or explain briefly why no such continuous function exists: (i) $I=[0,1], \quad J=[0, \infty)$; (ii) $I=(0,1], \quad J=[0, \infty)$; (iii) $I=(0,1], \quad J=(-\infty, \infty)$.