Definition:

A sequence of functions fn : S → X, where (X, d) is a metric space, is said to converge locally uniformly to f if for any x ∈ S there is a neighbourhood U of x such that fn|U converges uniformly to f|U.

Here f|U denotes the restriction of f to U, i.e. the function f|U : U → X given by f|U(x) = f(x) for x ∈ U. When it is clear from the context that we mean the restriction we often do not bother to write out the |U though.

Locally uniform convergence is handy, since even though we might not quite have uniform convergence the limit function f will still have nice properties guaranteed by uniform convergence. As an example we can prove a simple result on continuity of limits, and the same reasoning can be used to show that other local properties of uniform limits also hold for locally uniform limits.

If fn : S → X are continuous and converge locally uniformly to f then f is continuous.

Proof:

For any x ∈ S there is a neighbourhood U of x such that f converges uniformly to f on U. So on U f is a uniform limit of continuous functions and therefore f is continuous on U. In particular f is continuous at x.
Hence f is continuous at all x ∈ S, so f is continuous.

A particularly important example of locally uniform convergence is that any complex power series converges locally uniformly within its radius of convergence.

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