Let f : 0 , 3 → R be defined by f x = min x - x , 1 + x - x where x is the greatest integer less than or equal to x . Let P denote the set containing all x ∈ 0 , 3 where f is discontinuous, and Q denote the set containing all x ∈ 0 , 3 where f is not differentiable. Then the sum of number of elements in P and Q is equal to _____.

Step-by-step solution

Given, f x = min x - x , 1 + x - x So, the graph of the function is 1 - x = 1 - x ; 0 ≤ x < 1 From the graph we can say that the function is continuous at all points in the interval 0 , 3 . So, P = 0 Again, from the diagram, we can say that at x = 1 2 , 1 , 3 2 , 2 , 5 2 , there are sharp edges. So, the function is non differentiable at x = 1 2 , 1 , 3 2 , 2 , 5 2 Hence, Q = 5 So, P + Q = 5