On Uniform Convergence Page: 7
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3f (x)—* ^ en (0, 1), but Sf(x) doea not conwrge uniformly
n **
to sero on (0^, 1)*. For let e *= l/2 and ohoose any N*
n > N let x ss l/n;, thang,
)F*n(x) * ss tl/nx ^ 0! *= 1 > e*
Thus $11 of the eonditiona of the denial of uniform, convey*-
genes aye met.
1.16. As is suggested in l-t.3,5*. it may bo of interest
to give axplieitly the following-positive definition that.
Sfg^k)' does net oonverge tmiforml'y to the function f(x) on
[a,, b] (in notation aa indioated)i
Sf^(x)i4f(x) on [a., b] ;ssi. there exista an e > 0 such
th&t-,, for every N there exia.ts an n > N and # point in
(a,< b] such that^
n o o
§k. Aagusaptlons
1.1?. It will be assumed in this paper that the reader
knows the fundam#ntala of elementary Analysts and a. few
elementary set properties. Per axsspl#? it will bo aa#UBad
that th# reader is familiar with the concepts of continuity*
differentiability, Lebesgue Measure gero, And Riamsgm int&*
gration in the third ahapter* Also so^e knowledge of series
of numbers end ordinary aonvergenoa of series of function#
ia assumed. Listed below are some theorems without proofa
that will be referred to in this paper*
1*19. A neeeagary and sufficient condition that
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Drew, Dan Dale. On Uniform Convergence, thesis, February 1951; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc96930/m1/10/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .