Question

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

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

 

Answer 1

Given:

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

Putting x−a = y, we get

\[\lim_{x \to a} \left( x - a \right) \sin\left( \frac{1}{x - a} \right) = \lim_{y \to 0} y \sin\left( \frac{1}{y} \right)\]

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

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

Hence, f(x) is continuous at x = a.