On Uniform Convergence Page: 4
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4
special o&sea of a#quaag#s and aeries of functions where
the functions are oonstgnta. For a aeqmenea or a aerie#
of functions^ we must$p.e^ of a dOE&in of definition# i.e.,
the values of x for which f^(x) ia defined for n - 1? 2y, ... %
and & dosmin of conwrgenceA i.e. , the values of x for which
or Sf^(x) converge. In thia pap#p, w# will be con-
cerned primarily with series of functions and & partlcul&r
type of oenwrgenoe which is described b#low% The y^rp#g#
of this p&par is to familiarise the re&d#r with the eonoept
of uniform eonvergenee# In the main it ia & compilation of
aatwial found in various references and retised to eonfora
to standard notation*
$2* Definitions
1*8, In this paper the symbol ;w; signifies "i$ defined
to be", "m^ans th&t"$. or "wans"*. A doisain of definition
(of a, fwnotion^ aeries^ etc.) :^: a aet of real numbers (or
points)* We will., without loa# of ganarali&ationy state the
thaoross and definitions of this paper in terms of interval#
instead of a g^ner#! domain of definition*. A closed' inteyy.^1..,
in notation [a, b], consist# of the sot of points x such that,
a < x < b, An open interval* in notation (a.* b), consists
of the set of points x such that a ^ x <rb*- The function
f(x) ia bounded on b] there exists &-flnito mnaber K
such thatjf(x)j < K on [a, b]{ similarly y. f(x) ia bounded
from above on b] i^: there exists a finite K such that
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Drew, Dan Dale. On Uniform Convergence, thesis, February 1951; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc96930/m1/7/?rotate=0: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .