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Functions preserving limits of subsequences

Raluca-Ilinca B˘alilescu 1

1 Introduction

Given a bounded sequence x = (x n ) ⊆ R, let us denote by L (x) ⊆ R the set of all real
n≥1
numbers α for which there exists at least one subsequence of x converging to α. That is,

L (x) = {α ∈ R : ∃k 1 < k 2 < · · · , lim x k n = α}. (1)
n→+∞

It is well-known from basic real analysis that:

• for each bounded sequence x = (x n ) , we have that L (x) is a nonempty compact subset
n≥1
of the real line;

• given a bounded sequence x = (x n ) , the set L (x) is a singleton if and only if the
n≥1
sequence x converges to the unique element of L (x) ;

• for each bounded sequence x = (x n ) , we have that
n≥1

{lim inf(x), lim sup(x)} ⊆ L(x) ⊆ [lim inf(x), lim sup(x)], (2)

where, as usual, lim inf(x) = sup k≥1 (inf n≥k x n ) and lim sup(x) = inf k≥1 (sup n≥k x n ) are the
inferior (respectively, superior) limit of the sequence x.

Equality may occur in the left hand side inclusion or in the right hand side inclusion
of (2), even if lim inf(x) ̸= lim sup(x): if x n = 0 for odd n and x n = 1 for even n, then
lim inf(x) = 0, lim sup(x) = 1 and L (x) = {0, 1}, while if we enumerate [0, 1] ∩ Q as a sequence
x = (x n ) , then lim inf(x) = 0, lim sup(x) = 1 and L (x) = [0, 1].
n≥1
If x = (x n ) ⊆ R is not bounded, then L (x) becomes a subset of the extended real line
n≥1
R ∪ {±∞}: in this case it contains α ∈ R (if any) as in (1), and α ∈ {±∞} for which there
exists a subsequence of x which diverges to α. If x is unbounded, then L (x) ∩ R might be
empty, but L (x) ∩ {±∞} is not.
Consider now a function f : R → R. Given a sequence x = (x n ) ⊆ R, in what follows
n≥1
we shall denote by f (x) the sequence obtained by applying the function f to each term of
x, that is f (x) = (f(x n )) ⊆ R. A well-known result of real analysis states that functions
n≥1
f : R → R such that f (lim (x)) = lim (f (x)) for each convergent sequence x = (x n ) ⊆ R
n≥1
are precisely the continuous one. (We have denoted lim (x) = lim n→∞ x n for the convergent
1
Student, Faculty of Mathematics and Informatics, Ovidius University, Constant , a, raluca.balilescu@gmail.com

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