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 __________.

<p>$$\begin{aligned} f(x) & =\left[\frac{x}{3}+3\right]-[\sqrt{x}] \\ & =\left[\frac{x}{2}\right]-[\sqrt{x}]+3 \end{aligned}$$</p> <p>Critical points where $f(x)$ might change behaviours when $\frac{x}{2} \in$ integer and $\sqrt{x} \in$ integer</p> <p>$\Rightarrow$ Critical points,</p> <p>$$\begin{aligned} & f(0)=3 \\ & f\left(0^{+}\right)=3 \\ & f\left(1^{-}\right)=3 \\ & f\left(1^{+}\right)=2 \\ & f\left(2^{-}\right)=2 \\ & f\left(2^{+}\right)=3 \\ & f\left(3^{+}\right)=f\left(3^{-}\right)=3=f\left(4^{+}\right)=f\left(4^{-}\right)=f\left(5^{+}\right)=f\left(5^{-}\right) \\ & f\left(6^{-}\right)=3 \\ & f\left(6^{+}\right)=4 \\ & f\left(7^{-}\right)=f\left(7^{+}\right)=4 \\ & f\left(8^{-}\right)=4 \\ & f(8)=5 \end{aligned}$$</p> <p>$$\begin{aligned} & \Rightarrow f(x) \text { is not continuous at } \\ & x=1,2,6,8 \\ & \Rightarrow \sum_{a \in s} a=1+2+6+8 \\ & \quad=17 \end{aligned}$$</p>