If f(x) = begin{cases} x - 1, & x

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(C) To check the continuity of $f(x)$ at $x = 0$,we evaluate the left-hand limit,right-hand limit,and the value of the function at $x = 0$.$1$. Left-h

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

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(C) To check the continuity of $f(x)$ at $x = 0$,we evaluate the left-hand limit,right-hand limit,and the value of the function at $x = 0$.$1$. Left-hand limit $(LHL)$: $\mathop {\lim }\limits_{x 	o 0^-} f(x) = \mathop {\lim }\limits_{x 	o 0^-} (x - 1) = 0 - 1 = -1$.$2$. Right-hand limit $(RHL)$: $\mathop {\lim }\limits_{x 	o 0^+} f(x) = \mathop {\lim }\limits_{x 	o 0^+} (x^2) = 0^2 = 0$.$3$. Value of the function at $x = 0$: $f(0) = \frac{1}{4}$.Since $\mathop {\lim }\limits_{x 	o 0^-} f(x) 
eq \mathop {\lim }\limits_{x 	o 0^+} f(x)$,the limit $\mathop {\lim }\limits_{x 	o 0} f(x)$ does not exist.Therefore,$f(x)$ is discontinuous at $x = 0$.

If $f(x) = \begin{cases} x - 1, & x < 0 \\ \frac{1}{4}, & x = 0 \\ x^2, & x > 0 \end{cases}$,then

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