Theory of solar cells

Counterexample to a strengthening of the uniform limit theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions $\scriptstyle \scriptstyle \sin ^{n}(x)$ converge to the non-continuous red function. This can happen only if convergence is not uniform.

In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous.

Statement

More precisely, let X be a topological space, let Y be a metric space, and let ƒ_n : X → Y be a sequence of functions converging uniformly to a function ƒ : X → Y. According to the uniform limit theorem, if each of the functions ƒ_n is continuous, then the limit ƒ must be continuous as well.

This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒ_n : [0, 1] → R be the sequence of functions ƒ_n(x) = xⁿ. Then each function ƒ_n is continuous, but the sequence converges pointwise to the discontinuous function ƒ that is zero on [0, 1) but has ƒ(1) = 1. Another example is shown in the image to the right.

In terms of function spaces, the uniform limit theorem says that the space C(X, Y) of all continuous functions from a topological space X to a metric space Y is a closed subset of Y^X under the uniform metric. In the case where Y is complete, it follows that C(X, Y) is itself a complete metric space. In particular, if Y is a Banach space, then C(X, Y) is itself a Banach space under the uniform norm.

The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and ƒ_n : X → Y is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be uniformly continuous.

Proof

The uniform limit theorem is proved by the "ε/3 trick", and is the archetypal example of this trick.

More precisely we want to prove that, for every ε > 0, there is a neighbourhood N of any point x of X such that d_Y(f₀(x),f₀(y)) < ε whenever y is in N. In order to do so we make use of the triangle inequality applied to the metric d_Y on Y in the following way

$d_{Y}(f_{0}(x),f_{0}(y))\leq d_{Y}(f_{0}(x),f_{n}(x))+d_{Y}(f_{n}(x),f_{n}(y))+d_{Y}(f_{n}(y),f_{0}(y)).$

Let us fix an arbitrary ε > 0. Since the sequence of functions {f_n} converges uniformly to f by hypothesis then we may always choose a natural number n₀ such that n > n₀ implies

$d_{Y}(f_{n}(t),f_{0}(t))<{\frac {\epsilon }{3}},\qquad \forall t\in X.$

Moreover, since each f_n is continuous on the whole X by hypothesis, there is a neighbourhood N of any x such that d_Y(f_n(x),f_n(y)) < ε/3 whenever y is in N. Taking the limit on n to ∞ on the above triangular inequality we then obtain that, for every ε > 0, there is a neighbourhood N of any point x of X such that

$d_{Y}(f_{0}(x),f_{0}(y))\leq {\frac {\epsilon }{3}}+{\frac {\epsilon }{3}}+{\frac {\epsilon }{3}}=\epsilon$

whenever y is in N. Hence f is continuous everywhere on X by definition.

References

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