Which one of the following functions is not continuous on (0, )? - Vidyadip

MCQ

Which one of the following functions is not continuous on $(0,\pi )$?

A
$f(x)= cotx$
B
$g(x) =\int\limits_0^x {t\,\,\sin \frac{1}{t}\,\,dt} $
C
$h (x) = \left[ \begin{array}{l}1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\, < \,\,x\,\, \le \,\,\frac{{3\pi }}{4}\\\\2\sin \frac{2}{9}x\,\,\,\,\,\,\,\,\,\frac{{3\pi }}{4}\,\, < \,\,x < \,\,\pi \, \end{array} \right.$
✓
$l (x) = \left[ \begin{array}{l}x\sin x\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,0 < x \le \frac{\pi }{2}\,\\ \\\frac{\pi }{2}\,\sin (x + \pi )\,\,,\,\,\frac{\pi }{2}\, < \,x\, < \,\pi \end{array} \right.$

✓

Answer

Correct option: D.

$l (x) = \left[ \begin{array}{l}x\sin x\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,0 < x \le \frac{\pi }{2}\,\\ \\\frac{\pi }{2}\,\sin (x + \pi )\,\,,\,\,\frac{\pi }{2}\, < \,x\, < \,\pi \end{array} \right.$

d
$g (x) = \int\limits_0^x {t\,\,\,\sin \frac{1}{t}\,} \,\,dt$
$g (x) = x \,\sin(1/x)$ which is diff $\Rightarrow$ cont. in $(0, \pi )$
$l (x) =$ $\left[ \begin{array}{l}x\,\,\sin x\,\,\,\,\,\,\,\,\,\,\,\,\,0\, < \,x < \pi \,\,/2\\ - \pi /2\,\sin x\,\,\,\,\pi /2\, < \,x < \pi \end{array} \right.$
obvious discontinuity at $x = \pi /2$

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