Removable Discontinuity:

• Limit exists at x = a, but either f(a) is undefined or f(a) != lim(x->a) f(x)
• Can be "fixed" by redefining f(a) = lim(x->a) f(x)
• Example: f(x) = $\frac{x^2 - 1}{(x - 1)}$ at x = 1. Limit = 2, but f(1) is undefined.

Jump Discontinuity (First Kind):

• Both LHL and RHL exist but LHL != RHL
• The jump = |RHL - LHL|
• Example: f(x) = [x] (greatest integer function) at any integer n. LHL = n-1, RHL = n.

Infinite Discontinuity (Second Kind):

• At least one of LHL or RHL is +infinity or -infinity
• Example: f(x) = 1/x at x = 0. LHL = -infinity, RHL = +infinity.

Oscillatory Discontinuity:

• The function oscillates infinitely near x = a, so the limit does not exist
• Example: f(x) = sin$\frac{1}{x}$ at x = 0. Oscillates between -1 and 1 infinitely often.

JEE Tip: The greatest integer function [x] and fractional part {x} = x - [x] are favorite sources for discontinuity questions. [x] has jump discontinuity at every integer. {x} also has jump discontinuity at every integer (jumps from approaching 1 back to 0).