[x] is discontinuous at every integer. At integer n: LHL = n-1, RHL = n, f(n) = n. Jump of 1.

[f(x)] is discontinuous wherever f(x) is an integer. Find all x where f(x) = integer n, and check if the function crosses through n.

{x} = x - [x] is discontinuous at integers. At integer n: LHL of {x} = 1, RHL = 0. Jump from 1 to 0.

sin([x]) is discontinuous at every integer (since [x] jumps at integers).

[sin x] is discontinuous where sin x is an integer: sin x = 0 at x = npi, sin x = 1 at x = pi/2 + 2kpi, sin x = -1 at x = -pi/2 + 2k*pi.