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Keywords:
Woodcutters Problem; Baire one; Darboux
Summary:
Among the many characterizations of the class of Baire one, Darboux real-valued functions of one real variable, the 1907 characterization of Young and the 1997 characterization of Agronsky, Ceder, and Pearson are particularly intriguing in that they yield interesting classes of functions when interpreted in the two-variable setting. We examine the relationship between these two subclasses of the real-valued Baire one defined on the unit square.
References:
[1] Agronsky, S. J., Ceder, J. G., Pearson, T. L.: Some characterizations of Darboux Baire 1 functions. Real Anal. Exch. 23 (1997), 421-430. MR 1640015 | Zbl 0943.26004
[2] Bruckner, A. M.: Differentiation of Real Functions. CRM Monograph Series, Vol. 5. American Mathematical Society (AMS) (1994). MR 1274044
[3] Cavaretta, A. S., Dahmen, W., Micchelli, C. A.: Stationary Subdivision. Mem. Am. Math. Soc. 453 (1991). MR 1079033 | Zbl 0741.41009
[4] Evans, M. J., Humke, P. D.: A characterization of Baire one functions of two variables. J. Math. Anal. Appl. 335 (2007), 1-6. DOI 10.1016/j.jmaa.2007.01.034 | MR 2340300 | Zbl 1127.26003
[5] Evans, M. J., Humke, P. D.: Revisiting a century-old characterization of Baire class one, Darboux functions. Am. Math. Mon. 116 (2009), 451-455. DOI 10.4169/193009709X470344 | MR 2510842
[6] Evans, M. J., Humke, P. D.: Collections of Darboux-like, Baire one functions of two variables. (to appear) in J. Appl. Anal. MR 2680539
[7] Kuratowski, K.: Topology, Vol. I. Academic Press New York (1966). MR 0217751 | Zbl 0158.40901
[8] Malý, J.: The Darboux property for gradients. Real Anal. Exch. 22 (1996), 167-173. MR 1433604
[9] Micchelli, C. A., Prautzsch, H.: Uniform refinement of curves. Linear Algebra Appl. 114/115 (1989), 841-870. MR 0986909 | Zbl 0668.65011
[10] Young, W. H.: A theorem in the theory of functions of a real variable. Rend. Circ. Mat. Palermo 24 (1907), 187-192. DOI 10.1007/BF03015058
[11] Young, W. H.: A theorem in the theory of functions of a real variable. Rend. Circ. Mat. Palermo 24 (1907), 187-192. DOI 10.1007/BF03015058
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