Problem 107. Use the definition of continuity to show that Sa sin (4). ifo if -0 (r) - is continuous at 0.

Continuity: A function $f\left(x\right)$ is said to be continuous at $x=a$, if

i) $\underset{x\to a}{\lim }f\left(x\right)$ exists

ii) $f\left(a\right)$ exists and

iii) $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$

Given: $f\left(x\right)=\left\{\begin{array}{ll}x\sin \left(\frac{1}{x}\right),& \mathrm{if} x\ne 0\\ 0,& \mathrm{if} x=0\end{array}\right.$

To show: $f\left(x\right)$ is continuous at $x=0$ using definition of continuity.