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Question Number 25846 by rita1608 last updated on 15/Dec/17

$$determinetheintervalonwhichthe$$$$givenfunctioniscontinuous$$$$f\left(x\right)=\{\begin{array}{ll}\sin \frac{1}{x}& x\ne 0\\ x& x=0\end{array}$$

Commented by kaivan.ahmadi last updated on 15/Dec/17

$$\mathrm{since}\mathrm{li}\underset{\mathrm{x}\to 0}{\mathrm{m}sin}\frac{1}{\mathrm{x}}\ne \mathrm{f}\left(0\right),\mathrm{f}\left(\mathrm{x}\right)\mathrm{is}\mathrm{not}\mathrm{continuous}.$$$$\mathrm{this}\mathrm{limit}\mathrm{is}\mathrm{not}\mathrm{exixst}.$$$$\mathrm{so}\mathrm{the}\mathrm{interval}\mathrm{is}\mathbb{R}-\left\{0\right\}$$

Commented by prakash jain last updated on 16/Dec/17

Commented by kaivan.ahmadi last updated on 17/Dec/17

$$\mathrm{in}\mathrm{definition}\mathrm{we}\mathrm{must}\mathrm{have}$$$$\mid \sin \left(\frac{1}{\mathrm{x}}\right)-\mathrm{A}\mid <ϵ.$$$$$$$$\mathrm{why}\sin \frac{1}{\mathrm{x}}⩾ϵ?$$

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