Continuity and Uniform Continuity
This domain tests the ε-δ and sequential definitions of continuity, the global theorems available on compact intervals, and the distinction between continuity at each point and uniform continuity on a set. Expect both proofs and counterexamples on intervals that are open or unbounded.
Definitions and equivalences
f is continuous at c if for every ε > 0 there is δ > 0 such that |x − c| < δ implies |f(x) − f(c)| < ε. On a set, δ may depend on both c and ε. The sequential criterion says f is continuous at c iff x_n → c always implies f(x_n) → f(c). In R^n, continuity is equivalent to the preimage of every open set being open.
Global theorems
Extreme value theorem: a continuous function on a nonempty compact set is bounded and attains its maximum and minimum. Intermediate value theorem: on an interval, every value between f(a) and f(b) is attained. Uniform continuity requires that for every ε > 0 there is a single δ > 0 such that |x − y| < δ implies |f(x) − f(y)| < ε for all x, y in the domain. Heine–Cantor: continuity on a compact set implies uniform continuity. Lipschitz functions are uniformly continuous, but the converse fails, as with the square root on [0, 1].
Worked example
Show that f(x) = 1/x is not uniformly continuous on (0, 1). Take x_n = 1/n and y_n = 1/(n+1). Then |x_n − y_n| = 1/(n(n+1)) → 0, while |f(x_n) − f(y_n)| = |n − (n+1)| = 1. For ε = 1/2 no δ can work: given any δ > 0, choose n large enough that |x_n − y_n| < δ; the function values still differ by 1. The function is continuous on (0, 1); the failure is caused by the interval not being compact.
Traps
Letting δ depend on x in a purported uniform-continuity proof; concluding uniform continuity on an open interval from ordinary continuity; forgetting that the extreme value theorem needs compactness (x on R is continuous but unbounded); applying the intermediate value theorem on a disconnected domain; confusing continuity with differentiability.