IJPAM: Volume 79, No. 1 (2012)

DISCONTINUITY SETS OF SOME INTERVAL BIJECTIONS

E. de Amo$^1$, M. Dıaz Carrillo$^2$, J. Fernández Sánchez$^3$
$^{1,3}$Departamento de Álgebra y Análisis Matemático
Universidad de Almerıa
04120, Almerıa, SPAIN
$^2$Departamento de Análisis Matemático
Universidad de Granada
18071, Granada, SPAIN


Abstract. In this paper we prove that the discontinuity set of any bijective function from $\left[ a,b\right[ $ to $\left[ c,d\right] $ is infinite. We then proceed to give a gallery of examples which shows that this is the best we can expect, and that it is possible that any intermediate case exists.

We conclude with an example of a (nowhere continuous) bijective function which exhibits a dense graph in the rectangle $\left[ a,b\right[ \times \left[ c,d\right] .$

Received: April 19, 2012

AMS Subject Classification: 26A06, 26A09

Key Words and Phrases: discontinuity set, fixed point

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 79
Issue: 1