Uniform Continuity

One δ for the whole domain — a global strengthening of continuity.

The idea

Uniform continuity strengthens ordinary continuity by demanding one $\delta$ that serves the whole domain at once.

Ordinary continuity works point by point: given $x$ in the domain $D$ and a tolerance $\varepsilon > 0$, there is a $\delta > 0$ such that every point within $\delta$ of $x$ is sent within $\varepsilon$ of $f(x)$. The $\delta$ may depend on $x$, and usually does: where the graph is steep, $\delta$ must be small. The figure shows this dependence — the same tolerance $\varepsilon$ is met on a flat stretch of the graph by a window of wide half-width $\delta_{1}$, but on a steep stretch only by the far narrower $\delta_{2}$.

Definition (Uniform continuity).

A function $f$ is uniformly continuous on a set $D$ when $\forall \varepsilon > 0 \ \exists \delta > 0 \ \forall x, y \in D : \ |x - y| < \delta \implies |f(x) - f(y)| < \varepsilon.$

Ordinary continuity reads $\forall x \in D \ \forall \varepsilon > 0 \ \exists \delta > 0$. The quantifier $\forall x$ has moved. In the ordinary version it stands to the left of $\exists \delta$, so $\delta$ is chosen with the point already known and may be tailored to it. In the uniform version it stands to the right, so one $\delta$ must be chosen before the points are, and that choice must serve them all.

The order of the quantifiers is the entire difference, and it is why the verdict depends on the domain and not on the formula alone. On a closed bounded interval the two notions agree.

Theorem (Continuity on a closed interval).

A function continuous on a closed bounded interval $[a, b]$ is uniformly continuous on $[a, b]$.

Failure therefore requires an unbounded domain or a missing endpoint.

Ways to work on it

• Walkthrough. The single-δ definition, with an example that satisfies it and one that fails.
• Practice. Find the largest uniform δ for a linear function.
• Hardest. Decide whether one function is uniformly continuous on two different domains.

Not sure where to start? Take the ten-question placement test.