Greatest Integer Function [x]: The largest integer less than or equal to x.

• [3.7] = 3, [-2.3] = -3, [5] = 5
• Discontinuous at every integer (jump of 1)
• At integer n: LHL = n-1, RHL = n, f(n) = n

Fractional Part {x} = x - [x]: Always in [0, 1).

• Discontinuous at every integer (jumps from approaching 1 back to 0)
• At integer n: LHL = 1, RHL = 0, f(n) = 0

Key limits involving [x]:

• lim(x->n-) [x] = n - 1 (for integer n)
• lim(x->n+) [x] = n (for integer n)
• lim(x->0+) x[1/x] = 1 (using the fact that [1/x] is approximately 1/x - {1/x})

Continuity of [f(x)]: The composite [f(x)] is discontinuous wherever f(x) is an integer. So to find discontinuities, solve f(x) = n for integer values of n and check.

JEE Favorite: Find lim(x->0) [sin x / x]. Since sin x / x -> 1 from below (for small positive x, sin x < x so sin x / x < 1), [sin x / x] = 0 for small positive x. But from the left, sin x / x also approaches 1 from below, so [sin x / x] = 0. Thus the limit is 0, NOT 1.