Analysis I | Part IA, 2002

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 f1:(0,1)Rf_{1}:(0,1) \rightarrow \mathbb{R} which is not bounded.

(ii) A continuous function f2:(0,1)Rf_{2}:(0,1) \rightarrow \mathbb{R} which is bounded but does not attain its bounds.

(iii) A bounded function f3:[0,1]Rf_{3}:[0,1] \rightarrow \mathbb{R} which is not continuous.

(iv) A function f4:[0,1]Rf_{4}:[0,1] \rightarrow \mathbb{R} which is not bounded on any interval [a,b][a, b] with 0a<b1.0 \leqslant a<b \leqslant 1 .

[Hint: Consider first how to define f4f_{4} on the rationals.]

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