Continuity — Epsilon-Delta, Uniform Continuity

GRE Math Subject · Axiom Academy

LESSON Continuity — Epsilon-Delta, Uniform Continuity Advanced graduate-level mathematics content Step 1: Pointwise Continuity (ε-δ Definition) The rigor of analysis depends on formalizing what continuity means. The ε-δ definition makes this precise. Definition: A function is continuous at if: For every , there exists such that In words: f(x) can be made arbitrarily close to f(a) by making x sufficiently close to a. The δ depends on ε: typically δ = δ(ε, a) — the smaller ε, the more restrictive δ may need to be Continuity requires all three conditions: f(a) exists, lim x→a f(x) exists, and they're equal This is pointwise continuity—continuity at a single point Sometimes we need a stronger notion: continuity that is "uniform" across an entire domain. This is crucial for many theoretical results. Definition: is uniformly continuous on if: For every , there exists (depending on ε but not on x or y) such that Pointwise Continuity: "For each point a and each ε, there exists a δ(ε, a)" Uniform Continuity: "For each ε, there exists a δ(ε) that works for ALL points simultaneously" Uniform continuity is stronger: it implies pointwise continuity A continuous function on a compact interval is uniformly continuous (Heine-Cantor theorem) On non-compact domains, continuity ≠ uniform continuity (e.g., f(x) = 1/x on (0,1)) Step 3: Proving Continuity and Solving GRE Problems On the GRE, you'll need to prove continuity or identify discontinuities. Here's the systematic approach.

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