Question:

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f:[0,\infty)\to\mathbb{R}$ be a function defined by $f(x)=\left[\dfrac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $f$ is not continuous. Then $\sum_{a\in S}a$ is equal to ________.