Continuity

A function f(x) is continuous at

A function f(x) is continuous at point x = a if three conditions are met: (1) f(a) is defined, (2) lim(x→a) f(x) exists, (3) lim(x→a) f(x) = f(a). Exa

Right continuity

Right continuity: lim(x→a⁺) f(x) = f(a), Left continuity: lim(x→a⁻) f(x) = f(a). For piecewise functions like f(x) = x² for x ≤ 1, 2x for x > 1, check

Occurs when lim(x→a) f(x) exists but

Occurs when lim(x→a) f(x) exists but either f(a) is undefined or lim(x→a) f(x) ≠ f(a). Example: f(x) = (x²-4)/(x-2) for x≠2. At x=2, f is undefined bu

Occurs when both left and right

Occurs when both left and right limits exist but are different: lim(x→a⁻) f(x) ≠ lim(x→a⁺) f(x). Example: Greatest integer function [x] at x = 3 has l

Occurs when function approaches ±∞ as

Occurs when function approaches ±∞ as x approaches a point. Example: f(x) = 1/x at x = 0, where lim(x→0⁺) f(x) = +∞ and lim(x→0⁻) f(x) = -∞. Creates v