Limit, Continuity and Differentiability: JEE Main 2024 Mathematics question

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