Uniform Convergence

Pointwise vs uniform: when the limit of nice functions stays nice.

The idea

Uniform convergence is the notion of convergence for sequences of functions that is strong enough to carry continuity and integration to the limit. A sequence $f_{1}, f_{2}, f_{3}, \dots$ can approach a limit function $f$ in two senses.

The first works one point at a time: fix $x$ and require the sequence of numbers $f_{1}(x), f_{2}(x), \dots$ to converge to $f(x)$. Requiring this at every point gives pointwise convergence: $\forall x \ \forall \varepsilon > 0 \ \exists N \ \forall n \ge N : \ |f_{n}(x) - f(x)| < \varepsilon.$

Uniform convergence moves $\forall x$ all the way to the right: $\forall \varepsilon > 0 \ \exists N \ \forall n \ge N \ \forall x : \ |f_{n}(x) - f(x)| < \varepsilon.$ Moving the quantifier changes who chooses first. In the pointwise version $N$ is chosen with the point already known and may grow as $x$ varies; in the uniform version one $N$ must serve the whole domain at once.

The uniform condition has a compact restatement. Call $\sup_{x} |f_{n}(x) - f(x)|$ the sup-norm of the error: the worst gap anywhere. Convergence is uniform exactly when that one number tends to $0$, which is to say the graph of $f_{n}$ eventually lies inside a thin band around the graph of $f$ — the tube between $f - \varepsilon$ and $f + \varepsilon$ shown in the figure.

The uniform notion is the one that survives passage to the limit: a uniform limit of continuous functions is continuous, and a uniform limit may be moved inside an integral. Pointwise convergence guarantees neither.

Ways to work on it

• Walkthrough. Pointwise vs uniform convergence and the sup-norm of the error.
• Practice. Compute a sup-norm, or decide whether convergence is uniform.
• Hardest. Show a sequence can converge pointwise to a continuous limit and still not uniformly.

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