Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On some fundamental problems in cluster set theory

Author: J. A. Eidswick
Journal: Proc. Amer. Math. Soc. 39 (1973), 163-168
MSC: Primary 26A15; Secondary 30A72
MathSciNet review: 0313454
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An attempt to characterize the local behavior of arbitrary functions of two real variables in terms of their cluster sets along various approach curves leads to two main problems: (1) Finding a ``small'' family $ {\Gamma _0}$ of approach curves such that cluster sets along $ {\Gamma _0}$ determine total cluster sets; (2) Finding a ``large'' family of approach curves along which cluster sets can be preassigned. A nice solution of (1) is found and (2) is partially solved. Conjectures are made concerning a link between (2) and sets which are always of the first category.

References [Enhancements On Off] (What's this?)

  • [1] Rodolfo Bettazzi, Sui punti di discontinuita delle funzioni di variabile reale, Rend. Palermo 6 (1892), 173-195.
  • [2] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Math. and Math. Physics, no. 56, Cambridge Univ. Press, Cambridge, 1966. MR 38 #325. MR 0231999 (38:325)
  • [3] O. Haupt and G. Aumann, Differential- und Integralrechnung. Vol. 2, de Gruyter, Berlin, 1938, p. 61.
  • [4] E. W. Hobson, The theory of functions of a real variable. Vol. 1, 3rd ed., Dover, New York, 1957.
  • [5] K. Kuratowski, Topologie. Vol. 1, PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, 1966. MR 19, 873; MR 36 #840. MR 0217751 (36:840)
  • [6] Kiyoshi Noshiro, Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Heft 28, Springer-Verlag, Berlin, 1960. MR 24 #A3295. MR 0133464 (24:A3295)
  • [7] Arthur Rosenthal, On the continuity of functions of several variables, Math. Z. 63 (1955), 31-38. MR 17, 245. MR 0072198 (17:245c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A15, 30A72

Retrieve articles in all journals with MSC: 26A15, 30A72

Additional Information

Keywords: Cluster sets, approach curves, truncations, sets which are always of the first category
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society