If f(x) = begin{cases} x sin x, & 0

Question

(A) To check the continuity at $x = \frac{\pi}{2}$,we evaluate the left-hand limit $(LHL)$,right-hand limit $(RHL)$,and the function value.$1$. Left-h

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$f(x)$ is discontinuous at $x = \pi/2$

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(A) To check the continuity at $x = \frac{\pi}{2}$,we evaluate the left-hand limit $(LHL)$,right-hand limit $(RHL)$,and the function value.$1$. Left-hand limit $(LHL)$: $\lim_{x 	o \frac{\pi}{2}^-} f(x) = \lim_{x 	o \frac{\pi}{2}} (x \sin x) = \frac{\pi}{2} \sin(\frac{\pi}{2}) = \frac{\pi}{2} 	imes 1 = \frac{\pi}{2}$.$2$. Right-hand limit $(RHL)$: $\lim_{x 	o \frac{\pi}{2}^+} f(x) = \lim_{x 	o \frac{\pi}{2}} \frac{\pi}{2} \sin(\pi + x) = \frac{\pi}{2} \sin(\pi + \frac{\pi}{2}) = \frac{\pi}{2} \sin(\frac{3\pi}{2}) = \frac{\pi}{2} 	imes (-1) = -\frac{\pi}{2}$.$3$. Function value: $f(\frac{\pi}{2}) = \frac{\pi}{2} \sin(\frac{\pi}{2}) = \frac{\pi}{2}$.Since $\lim_{x 	o \frac{\pi}{2}^-} f(x) 
eq \lim_{x 	o \frac{\pi}{2}^+} f(x)$,the function $f(x)$ is discontinuous at $x = \frac{\pi}{2}$.

If $f(x) = \begin{cases} x \sin x, & 0 < x \le \frac{\pi}{2} \\ \frac{\pi}{2} \sin(\pi + x), & \frac{\pi}{2} < x < \pi \end{cases}$,then

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