to functions on an arc
Authors:
Hidefumi Katsuura and Kenneth R. Kellum
Journal:
Proc. Amer. Math. Soc. 101 (1987), 629633
MSC:
Primary 54C10; Secondary 54C30, 54F15
MathSciNet review:
911022
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Recently Jo W. Heath [6] has shown that any to function from an arc onto a Hausdorff space must have infinitely many discontinuities. Here we investigate extending Heath's result to to functions for . Examples show that in general Heath's theorem cannot be extended even for functions from an arc into itself. However, if is a to function from an arc onto an arc, then we prove that has infinitely many discontinuities.
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 [2]
 P. Civin, Twotoone mappings of manifolds, Duke Math. J. 10 (1943), 4957. MR 0008697 (5:47e)
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 P. Gilbert, toone mappings of linear graphs, Duke Math. J. 9 (1942), 475486. MR 0007106 (4:88b)
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 O. G. Harrold, The nonexistence of a certain type of continuous transformation, Duke Math. J. 5 (1939), 789793. MR 0001358 (1:223c)
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 O. G. Harrold, Exactly transformations on connected linear graphs, Amer. J. Math. 62 (1940), 823834. MR 0002554 (2:75c)
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 J. W. Heath, Every exactly to function on the reals has an infinite set of discontinuities, Proc. Amer. Math. Soc. 98 (1986), 369373. MR 854049 (87i:54031)
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 V. Martin and J. H. Roberts, Twotoone transformations on manifolds, Trans. Amer. Math. Soc. 49 (1941), 117. MR 0004129 (2:324d)
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 J. Mioduszewski, On twotoone continuous functions, Dissertationes Math. (Rozprawy Mat.) 24 (1961). MR 0145490 (26:3021)
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 S. B. Nadler, Jr. and L. E. Ward, Jr., Concerning exactly images of continua, Proc. Amer. Math. Soc. 87 (1983), 351354. MR 681847 (84c:54059)
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 J. H. Roberts, Twotoone transformations, Duke Math. J. 6 (1940), 256262. MR 0001923 (1:319d)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919870911022X
PII:
S 00029939(1987)0911022X
Keywords:
to function,
to function
Article copyright:
© Copyright 1987
American Mathematical Society
