Discuss the continuity of the function f(x) at the point x = 0, where


Discuss the continuity of the function f(x) at the point x = 0, where

$f(x)=\left\{\begin{array}{r}x, x>0 \\ 1, x=0 \\ -x, x<0\end{array}\right.$



$f(x)=\left\{\begin{array}{c}x, x>0 \\ 1, x=0 \\ -x, x<0\end{array}\right.$

(LHL at $x=0$ ) $=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0}-(-h)=0$

$(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} f(h)=\lim _{h \rightarrow 0}(h)=0$

And, $f(0)=1$

$\therefore \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x) \neq f(0)$

Hence, $f(x)$ is discontinuous at $x=0$.

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