Question

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

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

Answer 1

 Given:

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

\[\Rightarrow f\left( x \right) = \begin{cases}\left( x - a \right)\sin\left( \frac{1}{x - a} \right), x > 0 \\ \left( x + a \right)\sin\left( \frac{1}{x + a} \right), x < 0 \\ 0, x = a\end{cases}\] 

We observe

(LHL at x = a) = 

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

(RHL at x = a) = 

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

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

Hence, f(x) is continuous at x = a.