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 $f(x) = \left[\frac{x}{2} + 3\right] - \left[\sqrt{x}\right].$ 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 ________.

Answer 1

$\left\lfloor \frac{x}{2} + 3 \right\rfloor is discontinuous at x = 2, 4, 6, 8$
$\sqrt{x} \text{ is discontinuous at } x = 1, 4$
$F(x) \text{ is discontinuous at } x = 1, 2, 6, 8$
Summing the values:
$\sum a = 1 + 2 + 6 + 8 = 17$