Let f: [0, 3] → R be defined by f(x) = min\ x - [x], 1 + [x] - x\...

Given,

$f\left(x\right) = min\left\{x - \left[x\right], 1 + \left[x\right] - x\right\}$

So, the graph of the function is

$1 - \left\{x\right\} = 1 - x; 0 \le x < 1$

From the graph we can say that the function is continuous at all points in the interval $\left[0, 3\right] .$

So, $P = 0$

Again, from the diagram, we can say that at $x = \dfrac{1}{2}, 1, \dfrac{3}{2}, 2, \dfrac{5}{2},$ there are sharp edges.

So, the function is non differentiable at $x = \dfrac{1}{2}, 1, \dfrac{3}{2}, 2, \dfrac{5}{2}$

Hence, $Q = 5$

So, $P + Q = 5$