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New type continuities via Abel convergence.

Huseyin Cakalli1, Mehmet Albayrak2

  • 1Department of Mathematics, Maltepe University, Marmara Eğıtım Köyü, Maltepe, 34857 İstanbul, Turkey.

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Summary
This summary is machine-generated.

Abel continuity preserves Abel convergent sequences. The uniform limit of Abel continuous functions is also Abel continuous, forming a closed set within continuous functions.

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Area of Science:

  • Real Analysis
  • Topology

Background:

  • Continuity is a fundamental concept in real analysis.
  • Different types of continuity have distinct properties and implications.
  • Abel convergence is a specific mode of sequence convergence.

Purpose of the Study:

  • To define and investigate Abel continuity.
  • To explore the relationship between Abel continuity and other continuity types.
  • To determine the properties of sets of Abel continuous functions.

Main Methods:

  • Defining Abel continuity based on sequence preservation.
  • Analyzing properties of functions on subsets of real numbers (ℝ).
  • Investigating the behavior of uniform limits of function sequences.

Main Results:

  • A function is Abel continuous if it preserves Abel convergent sequences.
  • The uniform limit of a sequence of Abel continuous functions is Abel continuous.
  • The set of all Abel continuous functions constitutes a closed subset of the set of all continuous functions.

Conclusions:

  • Abel continuity is a well-defined concept with closure properties under uniform convergence.
  • The study contributes to understanding different modes of continuity in real analysis.
  • The findings highlight the structural properties of the space of Abel continuous functions.