The function f(x) = |x| + frac{|x|}{x} is | vedclass

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(C) The function is defined as $f(x) = |x| + \frac{|x|}{x}$.To check for continuity at $x = 0$,we evaluate the left-hand limit $(LHL)$ and right-hand

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discontinuous at the origin because $\frac{|x|}{x}$ is discontinuous there

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(C) The function is defined as $f(x) = |x| + \frac{|x|}{x}$.To check for continuity at $x = 0$,we evaluate the left-hand limit $(LHL)$ and right-hand limit $(RHL)$.For $x $	ext{LHL} = \lim_{x ightarrow 0^-} (-x - 1) = -1$.For $x > 0$,$|x| = x$,so $f(x) = x + \frac{x}{x} = x + 1$.$	ext{RHL} = \lim_{x ightarrow 0^+} (x + 1) = 1$.Since $	ext{LHL} 
eq 	ext{RHL}$,the function is discontinuous at $x = 0$.Note that $|x|$ is continuous at $x = 0$,but $\frac{|x|}{x}$ (the signum function) is discontinuous at $x = 0$.Therefore,the function is discontinuous at the origin because $\frac{|x|}{x}$ is discontinuous there.Thus,option $(c)$ is correct.

The function $f(x) = |x| + \frac{|x|}{x}$ is

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