Paper 1, Section II, 10D

Analysis I | Part IA, 2015

(a) For real numbers a,ba, b such that a<ba<b, let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be a continuous function. Prove that ff is bounded on [a,b][a, b], and that ff attains its supremum and infimum on [a,b][a, b].

(b) For xRx \in \mathbb{R}, define

g(x)={x12sin(1/sinx),xnπ0,x=nπ(nZ)g(x)=\left\{\begin{array}{ll} |x|^{\frac{1}{2}} \sin (1 / \sin x), & x \neq n \pi \\ 0, & x=n \pi \end{array} \quad(n \in \mathbb{Z})\right.

Find the set of points xRx \in \mathbb{R} at which g(x)g(x) is continuous.

Does gg attain its supremum on [0,π]?[0, \pi] ?

Does gg attain its supremum on [π,3π/2][\pi, 3 \pi / 2] ?

Justify your answers.

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