Question

The function $f(x)=\begin{cases}x .\sin \left(\frac{1}{x}\right), x \neq 0 \\ 0, \quad x=0\end{cases}$ then

• A. $f$ is continuous at $x=0$
• B. $f$ is continuous only at $x=0$
• C. $f$ is discontinuous only at $x=0$
• D. $f$ is discontinuous at infinite number of points

Answer 1

Answer: (A)

$\lim\limits _{ x \rightarrow 0} f ( x )=\lim \limits_{ x \rightarrow 0}\left(x \cdot \sin \left(\frac{1}{x}\right)\right)$
$=\lim\limits _{ x \rightarrow 0}( x ) \cdot \lim\limits _{ x \rightarrow 0}\left(\sin \left(\frac{1}{x}\right)\right)$
$=0$. ( a real number $)=0= f (0)$.