On a problem of Mazur from ‶The Scottish Book″ concerning second partial derivatives
Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 175-181
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
We comment on a problem of Mazur from ‶The Scottish Book″ concerning second partial derivatives. We prove that if a function $f(x,y)$ of real variables defined on a rectangle has continuous derivative with respect to $y$ and for almost all $y$ the function $F_y(x):=f'_y(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f''_{yx}$ exists. We construct a separately twice differentiable function whose partial derivative $f'_x$ is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.
Keywords:
comment problem mazur scottish book concerning second partial derivatives prove function real variables defined rectangle has continuous derivative respect almost function y has finite variation almost everywhere rectangle partial derivative exists construct separately twice differentiable function whose partial derivative discontinuous respect second variable set positive measure solves mazur problem negative
Affiliations des auteurs :
Volodymyr Mykhaylyuk 1 ; Anatolij Plichko 2
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author = {Volodymyr Mykhaylyuk and Anatolij Plichko},
title = {On a problem of {Mazur} from {‶The} {Scottish} {Book''} concerning second partial derivatives},
journal = {Colloquium Mathematicum},
pages = {175--181},
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volume = {141},
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year = {2015},
doi = {10.4064/cm141-2-3},
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Volodymyr Mykhaylyuk; Anatolij Plichko. On a problem of Mazur from ‶The Scottish Book″ concerning second partial derivatives. Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 175-181. doi: 10.4064/cm141-2-3
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