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(C) Given function is $f(x) = \begin{cases} x - \frac{|x|}{x}, & x  0 \ 1, & x = 0 \end{cases}$.To check continuity at

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$f(x)$ is continuous at $x = 0$

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(C) Given function is $f(x) = \begin{cases} x - \frac{|x|}{x}, & x  0 \ 1, & x = 0 \end{cases}$.To check continuity at $x = 0$,we evaluate the left-hand limit $(LHL)$,right-hand limit $(RHL)$,and the value of the function $f(0)$.$LHL = \lim_{x 	o 0^{-}} f(x) = \lim_{x 	o 0^{-}} \left(x - \frac{|x|}{x}ight)$.Since $x $LHL = \lim_{x 	o 0^{-}} (x - (-1)) = \lim_{x 	o 0^{-}} (x + 1) = 0 + 1 = 1$.$RHL = \lim_{x 	o 0^{+}} f(x) = \lim_{x 	o 0^{+}} \left(x + \frac{|x|}{x}ight)$.Since $x > 0$,$|x| = x$,so $\frac{|x|}{x} = 1$.$RHL = \lim_{x 	o 0^{+}} (x + 1) = 0 + 1 = 1$.Given $f(0) = 1$.Since $LHL = RHL = f(0) = 1$,the function $f(x)$ is continuous at $x = 0$.

If the function $f(x) = \begin{cases} x - \frac{|x|}{x}, & x < 0 \\ x + \frac{|x|}{x}, & x > 0 \\ 1, & x = 0 \end{cases}$,then which of the following is true?

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