Show that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds.

Write down examples of the following functions (no proof is required).

(i) A continuous function $f_{1}:(0,1) \rightarrow \mathbb{R}$ which is not bounded.

(ii) A continuous function $f_{2}:(0,1) \rightarrow \mathbb{R}$ which is bounded but does not attain its bounds.

(iii) A bounded function $f_{3}:[0,1] \rightarrow \mathbb{R}$ which is not continuous.

(iv) A function $f_{4}:[0,1] \rightarrow \mathbb{R}$ which is not bounded on any interval $[a, b]$ with $0 \leqslant a<b \leqslant 1 .$

[Hint: Consider first how to define $f_{4}$ on the rationals.]