If the function $f(x)$ is expressed as $f(x) = \frac{x-|x|}{x}$ then—

Explanation

Given $f(x) = \frac{x-|x|}{x}$ $f(x) = \begin{cases} \frac{x+x}{x} }{x} & x < 0 \\ \frac{x-x}{x} } & x > 0 \end{cases}$ $f(x) = \begin{cases} \frac{2x}{x} & x < 0 \\ 0 0 & x > 0 \end{cases}$ $f'(x) = \begin{cases} 0 & x < 0 \\ 0 0 & x > 0 \end{cases}$ Now, $\lim_{x \to 0^+} = \lim_{x \to 10} = 0$ We got constant value so the function is continuous for all values of $x$ except $x = 0$