The problem you've shared involves a function $f(x) = \left\lfloor \frac{x}{2} + 3 \right\rfloor - \left\lfloor \sqrt{x} \right\rfloor$, where $\left\lfloor t \right\rfloor$ denotes the greatest integer less than or equal to $t$. The task is to find the sum of all values in the set $S$, which contains the points in the interval $[0, 8]$ where the function $f(x)$ is not continuous.

Step-by-step approach:

1. Discontinuity of the greatest integer function:

   • The function $f(x)$ involves the floor functions $\left\lfloor \frac{x}{2} + 3 \right\rfloor$ and $\left\lfloor \sqrt{x} \right\rfloor$, both of which create discontinuities at points where the argument of the floor function is an integer.
   • The function $\left\lfloor t \right\rfloor$ is discontinuous at integer values of $t$. Therefore, we need to check for discontinuities when $\frac{x}{2} + 3$ and $\sqrt{x}$ are integers.

2. Identifying discontinuities in $\left\lfloor \frac{x}{2} + 3 \right\rfloor$:

   • The function $\frac{x}{2} + 3$ takes integer values at: $\frac{x}{2} + 3 = n \implies x = 2(n - 3)$ where $n$ is an integer. The relevant integers in the interval $[0, 8]$ lead to the values:
     • For $n = 3$, $x = 0$
     • For $n = 4$, $x = 2$
     • For $n = 5$, $x = 4$
     • For $n = 6$, $x = 6$
     • For $n = 7$, $x = 8$

3. Identifying discontinuities in $\left\lfloor \sqrt{x} \right\rfloor$:

   • The function $\sqrt{x}$ takes integer values when $x$ is a perfect square. In the interval $[0, 8]$, the perfect squares are: $x = 0, 1, 4$

4. Union of points of discontinuity:

   • The total set of points where either of the floor functions is discontinuous is the union of the sets: $S = \{0, 1, 2, 4, 6, 8\}$

5. Summation:

   • The sum of the elements in $S$ is: $0 + 1 + 2 + 4 + 6 + 8 = 21$

Thus, the sum $\sum_{a \in S} a$ is equal to 21.

Let me know if you'd like more details or have any questions! Here are some related questions:

1. How does the greatest integer function $\left\lfloor t \right\rfloor$ work in general?
2. What are the conditions for the discontinuity of a function involving floor functions?
3. Can this method of analyzing discontinuities be applied to other types of piecewise functions?
4. How does the floor function differ from the ceiling function in terms of discontinuity?
5. What role do perfect squares play in discontinuity analysis for functions involving square roots?

Tip: When analyzing the continuity of functions involving floor or ceiling functions, always focus on the points where the arguments of those functions hit integer values.