Continuity: Function f is continuous at x=a if: (1) f(a) exists; (2) lim(x→a) f(x) exists; (3) lim(x→a) f(x) = f(a). Continuous on interval if continuous at every point. Discontinuity types: (1) Removable: lim exists but ≠ f(a); (2) Jump: left/right limits different; (3) Infinite: vertical asymptote. Example: f(x) = (x²-1)/(x-1) is discontinuous at x=1 (removable), but equals 2 for x≠1. Properties: (1) Sum/product/quotient of continuous functions is continuous; (2) Inverse of continuous bijection is continuous. Solving: Check all three conditions. Identify discontinuity type from graph or algebraically. Exam tip: Polynomial functions always continuous. Rational functions continuous except where denominator = 0.