Answer
$0$
Work Step by Step
$f(2)$ is not defined. We find out if $\displaystyle \lim_{x\rightarrow 2}f(x)=L$ exists, and if it does, we can define the continuous extension of $f$ at $x=2$ as
$F(x)=\left\{\begin{array}{ll}
f(x) , & \text{if }x\neq 2\\
L, & \text{if }x=2
\end{array}\right.$
The limit at x=2 exists and equals 0 because
- as x approaches 2 from the left, f(x) approaches 0,
- as x approaches 2 from the right, f(x) approaches 0,
- both one-sided limits exist and are equal $\Rightarrow$ a limit at x=2 exists,
$L=\displaystyle \lim_{x\rightarrow 2}f(x)=0$
So, if we assign $f(2)=0$,
the extended function would be continuous at x=2.
