Answer
For $f$ to be continuous at $x=2$, $f(2)$ should be assigned value $0$.
Work Step by Step
To make $f$ continuous at $x=2$, $f(2)$ must acquire a value so that $\lim_{x\to2}f(x)=f(2)$.
That means we need to calculate $\lim_{x\to2}f(x)$ and assign that value to $f(2)$.
- As $x\to2^-$, since $x\lt2$, we employ the function $f(x)=-2x+4$
$$\lim_{x\to2^-}f(x)=\lim_{x\to2^-}(-2x+4)=-2\times2+4=0$$
- As $x\to2^+$, since $x\gt2$, we employ the function $f(x)=0$
$$\lim_{x\to2^+}f(x)=\lim_{x\to2^+}0=0$$
Therefore, $\lim_{x\to2}f(x)=\lim_{x\to2^+}f(x)=\lim_{x\to2^-}f(x)=0$
So, for $f$ to be continuous at $x=2$, $f(2)$ should be assigned value $0$ as well.
