Let f(x) = { (sin (x - [x]))/(x - [x]), , x ∈ ( - 2, - 1); max { 2x,3[|x|]}, , |x| < 1; 1, , otherwise
where [t] denotes greatest integer ≤ t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is :

Solution

Maths sir · 3 steps

First turn each piece into a plain formula, then check only the joints: x = -2, -1, 0 and 1.

1. Simplify the pieces. On (-2,-1), [x] = -2, so x - [x] = x + 2 = t with t ∈ (0,1) and f = (sin t)/t. On |x| < 1, [|x|] = 0, so f = max{2x, 0}: this is 0 on (-1, 0] and 2x on [0, 1). Everywhere else, including x = -1, f = 1.
2. Where to look. Each piece is smooth on its own interval, so trouble can only come at x = -2, -1, 0, 1. Near x = -2, use (sin t)/t = 1 - t²/6 + …. It tends to 1 and its slope tends to 0 as t → 0⁺.

   Point	Left side	Right side	Continuous?	Differentiable?
   x = -2	value 1, slope 0	limit 1, slope 0	yes	yes
   x = -1	limit sin 1 ≈ 0.84	limit 0, while f(-1) = 1	no	no
   x = 0	value 0, slope 0	value 0, slope 2	yes	no
   x = 1	limit 2	value 1	no	no

3. So. f is discontinuous at x = -1 and x = 1, so m = 2. It is not differentiable at x = -1, 0, 1, so n = 3. The ordered pair is (2, 3).

The idea to keep. With piecewise functions, first rewrite every greatest-integer bracket as a constant on its interval. Then test only the joints, checking the limit first and the slope second.

Answer from the NTA final answer key. Solution by PrepWiser.