Discontinuity types: (1) Removable (hole): lim f(x) exists, equals some value L, but f(a) ≠ L or undefined. Fix by redefining f(a) = L. Example: f(x) = (x²-4)/(x-2) has removable discontinuity at x=2; limit = 4. (2) Jump: Left and right limits exist but different. Cannot be removed. Example: f(x) = floor(x) has jump at integers. (3) Infinite (pole): Function approaches ±∞. Occurs at vertical asymptotes. Example: f(x) = 1/(x-1) at x=1. (4) Oscillatory: Function oscillates without limit. Example: sin(1/x) as x→0. Identifying: Graph inspection, limit evaluation, algebraic analysis. Exam tip: Understand which discontinuities are removable. Recognize asymptotes geometrically and algebraically.