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# Limits and Continuity of Piecewise Function

APCALC-4IZYRG

Consider the following piecewise function:

$$f(x)=\left\{ \begin{matrix} 2{ x }^{ 2 } & x<0 \\\ 2 & x=0 \\\ 2\sin{x} & x>0 \end{matrix} \right.$$

...which of the following pairs of statements about the continuity of the piecewise function is true?

A

$f(x)$ is continuous at all points in the domain.
The limit exists for each point of $f(x)$ in the domain.

B

$f(x)$ is continuous at all points on $(-1,0)$.
Since there is a removable discontinuity at $x=0$, the limit does not exist for each point of $f(x)$ in the domain.

C

$f(x)$ is continuous at all points on $(-1,1)$.
The limit exists for each point of $f(x)$ in the domain.

D

$f(x)$ is continuous at all points on $(0,1)$.
The limit exists for each point of $f(x)$ in the domain.