Define continuity precisely and classify types of discontinuities.
A function f is continuous at x = a if three conditions hold: f(a) is defined, lim_{x→a} f(x) exists, and lim_{x→a} f(x) = f(a). Intuitively, the graph has no breaks, holes, or jumps at x = a.
Types of Discontinuities
Intermediate Value Theorem (IVT): if f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists c ∈ (a, b) with f(c) = k.
Show that x³ − x − 1 = 0 has a root in [1, 2]. f(1) = 1−1−1 = −1 < 0. f(2) = 8−2−1 = 5 > 0. By IVT, since f is continuous and changes sign, there is at least one root in (1, 2).
Remember This!
All polynomial, rational (at non-zero denominators), exponential, logarithmic, and trigonometric functions are continuous on their natural domains. Discontinuities only arise at domain restrictions.
Which type of discontinuity occurs at x = 2 for f(x) = (x² − 4)/(x − 2)?