Paper 1, Section II, 10D

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

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

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

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

Does $g$ attain its supremum on $[0, \pi] ?$

Does $g$ attain its supremum on $[\pi, 3 \pi / 2]$ ?

Justify your answers.

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