Define what it means for a subset $E$ of $\mathbb{R}^{n}$ to be convex. Which of the following statements about a convex set $E$ in $\mathbb{R}^{n}$ (with the usual norm) are always true, and which are sometimes false? Give proofs or counterexamples as appropriate.

(i) The closure of $E$ is convex.

(ii) The interior of $E$ is convex.

(iii) If $\alpha: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is linear, then $\alpha(E)$ is convex.

(iv) If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous, then $f(E)$ is convex.