Question

Discuss the continuity of the following functions at the indicated point(s): 

(ii) \[f\left( x \right) = \left\{ \begin{array}{l}x^2 \sin\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]

Answer 1

Given:

\[f\left( x \right) = \binom{ x^2 \sin\frac{1}{x}, x \neq 0}{0, x = 0}\]

We observe

\[\lim_{x \to 0} x^2 \sin\left( \frac{1}{x} \right) = \lim_{x \to 0} x^2 \lim_{x \to 0} \sin\left( \frac{1}{x} \right) = 0 \times \lim_{x \to 0} \sin\left( \frac{1}{x} \right) = 0\]

\[\Rightarrow \lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]

Hence, f(x) is continuous at x = 0.