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sequence x = (x n ) .) What about functions preserving limit inferior and limit superior of
n≥1
bounded sequences? This question was split into two parts in the article [1]:

(P 1 ): What are the functions f : R → R such that

f(lim sup(x)) ≤ lim sup(f(x)) (3)

for each bounded sequence x = (x n ) ?
n≥1
(P 2 ): What are the functions f : R → R such that

lim sup(f(x)) ≤ f(lim sup(x)) (4)

for each bounded sequence x = (x n ) ?
n≥1
(Of course, analogous questions may be asked for the limit inferior as well, but their answers
will follow immediately once the answers to (P 1 ) and (P 2 ) are obtained.) It is proved at [1,
Theorem 3] that functions f : R → R solving (P 1 ) are exactly the lower semi-continuous ones,
and at [1, Theorem 1] that a function f : R → R is a solution to (P 2 ) if and only if it is
increasing and right-continuous. Combining these two results, we see that a function f : R → R
satisfies
f(lim sup(x)) = lim sup(f(x)) (5)

for each bounded sequence x = (x n ) if and only if it is continuous and increasing [1, Theorem
n≥1
6].

2 Results

Observe that f(lim sup(x)) belongs to f (L (x)) for each bounded sequence x. If f (x) is also
bounded above, then the real number lim sup(f(x)) belongs to L (f(x)). Thus (5) in particular
implies that
f (L (x)) ∩ L (f (x)) ̸= Ø (6)

for each bounded sequence x = (x n ) . (Let us observe that (6) also holds if we suppose
n≥1
that f(lim inf(x)) = lim inf(f(x)) for each bounded sequence x, instead of (5).) What are the
functions f : R → R such that (6) holds for each bounded sequence x = (x n ) ⊆ R? That
n≥1
is, what are the functions f : R → R such that for each bounded sequence x there exist real
numbers α ∈ L (x) and β ∈ L (f (x)) such that f (α) = β? As in [1], we shall split this question
into two parts. First, observe that if (3) holds, there exist then a real number α ∈ L (x) and
β ∈ L (f (x)) such that f (α) ≤ β. Also, if (4) holds, there exist then a real number α ∈ L (x)
and β ∈ L (f (x)) such that f (α) ≥ β.
Theorem 1. Let f : R → R be a function. Then f is lower semi-continuous if and only if it
has the property that given any bounded sequence of real numbers x, there exist a real number
α ∈ L (x) and β ∈ L (f (x)) such that f (α) ≤ β.

Proof. One implication comes immediately from (3) and the remark preceding the statement
of the theorem. If f is lower semi-continuous, by [1, Theorem 3] we have that (3) holds. Since
lim sup(x) ∈ L (x) and lim sup f(x) ∈ L(f(x)), there exist the real number α = lim sup(x) ∈
L (x) and β = lim sup f(x) ∈ L(f(x)) such that f (α) ≤ β.