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# Let $f(x)= \left\{ \begin{array}{1 1} |x|\cos \frac{1}{x} & \quad if\;x\neq0 \\ 0 & \quad if\;x=0 \end{array} \right.$ then discuss the continuity of f(x) at x = 0

Toolbox:
• For continuity at '0' LHL = RHL = f(0)
• $\lim\limits_{x \to 0} \: \large\frac{tan^{-1}x}{x}=1$
LHL = RHL = $\lim\limits_{x \to 0} |x|\: cos\large\frac{1}{x}$
$cos\large\frac{1}{x} = K \: where \: -1 \leq K \leq 1$ for any $x \neq 0$

$\Rightarrow \lim\limits_{x \to 0} \: K| x |=0$
$f(0)=0 \Rightarrow f$ is continuous at x = 0

edited Mar 26, 2013