Answer
Prove that $\lim\limits_{x\to a}f(x)=f(a)$ for $\forall a\in[4,\infty)$.
Work Step by Step
*NOTES TO REMEMBER: $f(x)$ is continuous on the interval if and only if it is continuous at every point in the interval.
In other words, $f(x)$ is continuous on the interval $(u,v)$ if and only if for $\forall a\in(u,v)$, we have $$\lim\limits_{x\to a}f(x)=f(a)$$
For $\forall a\in[4,\infty)$, we consider
$\lim\limits_{x\to a}f(x)$
$=\lim\limits_{x\to a}(x+\sqrt{x-4})$
$=\lim\limits_{x\to a}x+\lim\limits_{x\to a}\sqrt{x-4}$
$=\lim\limits_{x\to a}x+\sqrt{\lim\limits_{x\to a}(x-4)}$
$=\lim\limits_{x\to a}x+\sqrt{\lim\limits_{x\to a}x-\lim\limits_{x\to a}4}$
$=a+\sqrt{a-4}$
$=f(a)$
Therefore, $f(x)$ is continuous on the interval $[4,\infty)$.
