A real-valued function
f of a real variable is said to be
-slowly varying (
-s.v.) if lim
x
(x) [f(x+
)–f(x)]=0 for each
. It is said to be uniformly
-slowly varying (u.
-s.v.) if lim
x sup
I
(x) |f(x+)–f(x)|=0 for every bounded interval
I.
It is supposed throughout that
is positive and increasing. It is proved that if
increases rapidly enough, then every
-s.v. function
f must be u.
-s.v. and must tend to a limit at
. Regardless of the rate of increase of
, a measurable function
f must be u.
-s.v. if it is
-s.v. Examples of pairs (
,f) are given that illustrate the necessity for the requirements on
and
f in these results.
The research of the first author was partially supported by NSF Grant # GP 14986.
The research of the third author was partially supported by a grant from the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant # AF OSR 68 1499.