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Discontinuous Functions with the Darboux Property

Published online by Cambridge University Press:  20 November 2018

Israel Halperin*
Affiliation:
Queen's University
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If f(x) is real-valued and continuous, it has the property that it takes on all intermediate values when it passes from one value to another. This means that whenever f(x1) and f(x2) are different and u is any number between them, then f(x) = u for at least one x between x1 and x2. We shall call this the Darboux property.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1959

References

1) Darboux, G., Mémoire sur les fonctions discontinues, Annales Scientifiques de l' Ecole Normale Supérieure, 2e série, 4 (1875), 57-112.CrossRefGoogle Scholar

2) Volterra, V., Giornale de Battaglini, 1881.Google Scholar

3) Lebesgue, H., Leçons sur l'intégration, (Paris, 1904).Google Scholar

4) Halperin, Israel, Discontinuous functions with the Darboux property, American Mathematical Monthly 57 (1950), 539-540.Google Scholar

5) Halperin, Israel, On the Darboux property,Pacific Journal of Mathematics 5 (1955), 703-705.CrossRefGoogle Scholar

6) Lebesgue, loc. cit., p. 85.

7) Darboux, loc. cit., p. 109.

8) For this type of construction in n-dimensional space, see Hahn and Rosenthal, Set Functions, University of New Mexico Press, 1948, p. 98, Theorem 8. 2. 8.

9) See Lebesgue, loc. cit., p. 92.

10) Lebesgue, loc. cit., p. 90. This example is cited in L.M. Graves, The Theory of Functions of Real Variables, McGraw-Hill, 1946, p. 65. Lebesgue uses this example to show that the sum of two functions need not have the Darboux property though each of the functions has it.

11) Hamel, G., Eine Basis aller Zahlen und die unstetigen Lësungen der Funktionalgleichung: f(x + y) = f(x) + f(y), Mathematische Annalen 60 (1905), 459-462.CrossRefGoogle Scholar

12) Ostrowski, A., Uber die Funktionalgleichung der Exponential - funktion und verwandte Funktionalgleichungen, Jahresbericht der Deutschen Mathematiker Vereinigung 38 (1929).Google Scholar

13) This construction was used by W. Sierpinski and N. Lusin to subdivide an interval into continuum many parts each of exterior Lebesgue measure equal to the length of the interval. See their paper in Comptes Rendus (Paris) 165 (1917), 422-424.