Slowly Oscillating Continuity

H. Çakalli

Abstract and Applied Analysis, 2008, vol. 2008, 1-5

Abstract:

A function ð ‘“ is continuous if and only if, for each point ð ‘¥ 0 in the domain, l i m ð ‘› → ∞ ð ‘“ ( ð ‘¥ ð ‘› ) = ð ‘“ ( ð ‘¥ 0 ) , whenever l i m ð ‘› → ∞ ð ‘¥ ð ‘› = ð ‘¥ 0 . This is equivalent to the statement that ( ð ‘“ ( ð ‘¥ ð ‘› ) ) is a convergent sequence whenever ( ð ‘¥ ð ‘› ) is convergent. The concept of slowly oscillating continuity is defined in the sense that a function ð ‘“ is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, ( ð ‘“ ( ð ‘¥ ð ‘› ) ) is slowly oscillating whenever ( ð ‘¥ ð ‘› ) is slowly oscillating. A sequence ( ð ‘¥ ð ‘› ) of points in ð ‘ is slowly oscillating if l i m 𠜆 → 1 + l i m ð ‘› m a x ð ‘› + 1 ≤ 𠑘 ≤ [ 𠜆 ð ‘› ] | ð ‘¥ 𠑘 − ð ‘¥ ð ‘› | = 0 , where [ 𠜆 ð ‘› ] denotes the integer part of 𠜆 ð ‘› . Using 𠜀 > 0 's and ð ›¿ 's, this is equivalent to the case when, for any given 𠜀 > 0 , there exist ð ›¿ = ð ›¿ ( 𠜀 ) > 0 and ð ‘ = ð ‘ ( 𠜀 ) such that | ð ‘¥ ð ‘š − ð ‘¥ ð ‘› | < 𠜀 if ð ‘› ≥ ð ‘ ( 𠜀 ) and ð ‘› ≤ ð ‘š ≤ ( 1 + ð ›¿ ) ð ‘› . A new type compactness is also defined and some new results related to compactness are obtained.

Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:485706

DOI: 10.1155/2008/485706

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