Continuity and Differentiability

Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x)=[x]+|x-2|, -2<x<3, is not continuous and not differentiable. Then m+n is equal to :

1. 6
2. 9
3. 8
4. 7

Answer

(3) 8

Solution

|x-2| is continuous; only [x] jumps, at the integers -1,0,1,2 of (-2,3).

[figure]

f is discontinuous at 4 points m=4

A jump point is never differentiable; the corner of |x-2| at x=2 is already counted n=4

m+n=8

Answer: (3)