Continuity (ε-δ Definition)

What ε-δ actually says

Forget graphs for a moment. Read the definition as a two-player game. A challenger demands that f(x) stay within ε of f(c) — they pick ε as small as they like. Your job is to respond with a δ small enough that, as long as the input x is within δ of c, the output f(x) is within ε of f(c). If you can always answer the challenger, no matter how tight the ε, the function is continuous at c.

Symbolically, f is continuous at c if

∀ ε > 0, ∃ δ > 0 such that |x − c| < δ ⇒ |f(x) − f(c)| < ε.

Three things are doing the work in that sentence. First, ε is universal — the challenger picks any ε. Second, δ is existential — you only need to find some δ, not the largest possible one. Third, δ is allowed to depend on ε (and on c); a smaller ε will usually force a smaller δ.

The intuitive picture: draw a horizontal band of vertical width 2ε centered on f(c). Continuity says you can always find a vertical band of width 2δ around c such that the graph stays inside the horizontal band whenever x stays inside the vertical one.

Worked example — proving f(x) = x² is continuous at x = 2

We want to show: for every ε > 0, there exists δ > 0 such that |x − 2| < δ implies |x² − 4| < ε.

Scratch work. Factor: |x² − 4| = |x − 2|·|x + 2|. The factor |x − 2| is what we control through δ. The factor |x + 2| we have to bound. If we pre-commit to δ ≤ 1, then |x − 2| < 1 means 1 < x < 3, so |x + 2| < 5. Then |x² − 4| < 5·|x − 2|. To force this below ε, take |x − 2| < ε/5.

Clean proof. Given ε > 0, choose δ = min(1, ε/5). Suppose |x − 2| < δ. Then |x − 2| < 1, so |x + 2| < 5. Therefore

|x² − 4| = |x − 2|·|x + 2| < (ε/5) · 5 = ε. ∎

Notice the structure: scratch work first (find δ that works), then a clean forward proof (verify it works). Almost every ε-δ proof you write follows this two-step pattern.

The three conditions for continuity at a point

A function f is continuous at c if and only if all three of the following hold:

1. f(c) is defined. The point c is in the domain.
2. The two-sided limit lim_x→c f(x) exists.
3. The limit equals the value: lim_x→c f(x) = f(c).

Each condition can fail independently, producing different "discontinuity zoos":

• Removable. The limit exists but f(c) is missing or wrongly defined. Example: g(x) = sin(x)/x at x = 0. Limit is 1 but g(0) is undefined. "Plug the hole."
• Jump. One-sided limits exist but disagree, so the two-sided limit does not exist. Example: the sign function at 0.
• Infinite. One or both sides blow up. Example: 1/x at 0.
• Essential / oscillatory. No reasonable limiting behaviour at all. Example: sin(1/x) at 0 oscillates infinitely fast.

Uniform continuity vs pointwise continuity

There are several flavours of continuity. The differences are in where the quantifiers live.

	Pointwise continuity	Uniform continuity
Quantifier order	∀ c ∀ ε ∃ δ	∀ ε ∃ δ ∀ c
Does δ depend on c?	Yes — δ may shrink as c moves	No — one δ works for the whole domain
Strength	Weaker	Stronger (implies pointwise)
Example that holds	x² on all of ℝ	x² on [0, 1]
Example that fails	1/x on (0, 1] is pointwise continuous but…	…not uniformly continuous (δ → 0 as x → 0)
Compactness theorem	—	Heine–Cantor: continuous on a compact set ⇒ uniformly continuous
Preserves Cauchy sequences	Not always	Always

The Heine–Cantor theorem is one of the cleanest payoffs of the ε-δ machinery: closed bounded intervals are compact, and on compact sets, plain continuity automatically upgrades to uniform continuity. That fact is what makes the Riemann integral work.

The topological reformulation

Once you are comfortable with ε-δ, you can throw away distances entirely. A function f: X → Y between topological spaces is continuous if and only if

For every open U ⊆ Y, the preimage f⁻¹(U) is open in X.

This is equivalent to ε-δ in metric spaces (the open sets there are exactly the unions of open balls), but it generalises immediately to settings where there is no notion of distance — quotient spaces, Zariski topologies in algebraic geometry, weak topologies in functional analysis. Most of modern analysis and topology runs on this version.

Classical applications

• Intermediate Value Theorem. A continuous function on [a, b] hits every value between f(a) and f(b). Used to prove that every odd-degree real polynomial has a real root.
• Extreme Value Theorem. A continuous function on a closed bounded interval attains its maximum and minimum. The starting gun for optimisation.
• Fixed point theorems. Brouwer's theorem: every continuous self-map of a closed disk has a fixed point. Used in game theory (Nash equilibrium) and economics (general equilibrium).
• Riemann integration. Continuous functions on [a, b] are Riemann integrable; the proof uses uniform continuity from Heine–Cantor.
• Compactness arguments. Continuous image of a compact set is compact — the engine behind countless existence proofs in PDEs and calculus of variations.

Common mistakes

• Quantifier order reversed. Writing "∃ δ ∀ ε" instead of "∀ ε ∃ δ" breaks the definition. The first version says one fixed δ works for all ε — that would force f(x) = f(c) on a whole neighbourhood, which is much stronger than continuity.
• Picking δ that depends on x. δ may depend on ε and c, but not on x. x is the variable being constrained.
• Confusing continuity with differentiability. Continuous does not imply differentiable. The Weierstrass function is the canonical counterexample: continuous everywhere, differentiable nowhere.
• Confusing "limit exists" with "continuous". sin(x)/x has a limit at 0 but is not defined there, so it is not continuous at 0 until you patch the value.
• Treating uniform and pointwise continuity as the same. They agree on compact sets but diverge sharply on unbounded or open domains. f(x) = 1/x on (0, ∞) is pointwise but not uniformly continuous.
• Skipping the scratch work. A clean ε-δ proof comes from working backward to discover δ, then writing the forward proof. Trying to find δ in one pass leads to circular arguments.