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(D) To check the continuity at $x = 0$: $1$. $f(0) = 1$. $2$. Left-hand limit: $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} (2[x] - \frac{x}{|x|}) = 2(-1

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right continuous at $x = 1$

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(D) To check the continuity at $x = 0$: $1$. $f(0) = 1$. $2$. Left-hand limit: $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^-} (2[x] - \frac{x}{|x|}) = 2(-1) - (-1) = -2 + 1 = -1$. $3$. Right-hand limit: $\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} (2[x] - \frac{x}{|x|}) = 2(0) - (1) = -1$. Since $\lim_{x 	o 0} f(x) = -1 
eq f(0)$, $f$ is discontinuous at $x = 0$. To check the continuity at $x = 1$: $1$. $f(1) = 2[1] - \frac{1}{|1|} = 2(1) - 1 = 1$. $2$. Left-hand limit: $\lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^-} (2[x] - \frac{x}{|x|}) = 2(0) - 1 = -1$. $3$. Right-hand limit: $\lim_{x 	o 1^+} f(x) = \lim_{x 	o 1^+} (2[x] - \frac{x}{|x|}) = 2(1) - 1 = 1$. Since $\lim_{x 	o 1^+} f(x) = f(1)$, the function is right continuous at $x = 1$.

If $[x]$ is the greatest integer function and $f(x) = \begin{cases} 2[x] - \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}$ is a real-valued function, then $f$ is

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