Consider the space of all bounded continuous functions
, with the uniform norm
. Let
be its subset that consists of all periodic continuous functions: recall that
is periodic if there exists
such that
for all
.
The set is not closed in the topology of
. Indeed, let
be the distance from
to nearest integer. The function
is periodic with
. Therefore, each sum of the form
is periodic with
. Hence the sum of the infinite series
is a uniform limit of periodic functions. Yet,
is not periodic, because
and
for
(for every
there exists
such that
is not an integer).

The above example (which was suggested to me by Yantao Wu) is somewhat similar to the Takagi function, which differs from it by the minus sign in the exponent: . Of course, the Takagi function is periodic with period
.

Do we really need an infinite series to get such an example? In other words, does the set contain an elementary function?
A natural candidate is the sum of trigonometric waves with incommensurable periods (that is, the ratio of periods must be irrational). For example, consider the function whose graph is shown below.

Since and
for all
, the function
is not periodic. Its graph looks vaguely similar to the graph of
. Is
a uniform limit of periodic functions?
Suppose is a
-periodic function such that
. Then
, hence
for all
, hence
. By the definition of
this implies
and
for all
. The following lemma shows a contradiction between these properties.
Lemma. If a real number satisfies
for all
, then
is an integer multiple of
.
Proof. Suppose is not an integer multiple of
. We can assume
without loss of generality, because
can be replaced by
to get it in the interval
and then by
to get it in
. Since
, we have
. Let
be the smallest positive integer such that
. The minimality of
implies
, hence
. But then
, a contradiction.
The constant in the lemma is best possible, since
for all
.
Returning to the paragraph before the lemma, choose so that
. The lemma says that both
and
must be integer multiples of
, which is impossible since they are incommensurable. This contradiction shows that
for any periodic function
, hence
is not a uniform limit of periodic functions.
The above result can be stated as . I guess
is actually
. It cannot be greater than
since
for all
. (Update: Yantao pointed out that the density of irrational rotations implies the distance is indeed equal to 1.)
Note: the set is a proper subset of the set of (Bohr / Bochner / uniform) almost periodic functions (as Yemon Choi pointed out in a comment). The latter is a linear space while
is not. I was confused by the sentence “Bohr defined the uniformly almost-periodic functions as the closure of the trigonometric polynomials with respect to the uniform norm” on Wikipedia. To me, a trigonometric polynomial is a periodic function of particular kind. What Bohr called Exponentialpolynom is a finite sum of the form
where
can be any real numbers. To summarize: the set considered above is the closure of
while the set of almost periodic functions is the closed linear span of
. The function
is an example of the latter, not of the former.