Measurability of partial derivatives
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- by Moshe Marcus and Victor J. Mizel PDF
- Proc. Amer. Math. Soc. 63 (1977), 236-238 Request permission
Abstract:
Let f be a real function defined in ${R_n}$. In this note we give a sufficient condition in order that the set of points where the partial derivative $\partial f/\partial {x_i}$ exists is Lebesgue measurable and $\partial f/\partial {x_i}$ is a measurable function on this set. This result unifies and extends a number of previous results.References
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U. S. Haslam-Jones, Derivative planes and tangent planes of a measurable function, Quart. J. Math. Oxford 3 (1932), 120-132.
- James Serrin, On the differentiability of functions of several variables, Arch. Rational Mech. Anal. 7 (1961), 359–372. MR 139700, DOI 10.1007/BF00250769
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 63 (1977), 236-238
- MSC: Primary 26A54
- DOI: https://doi.org/10.1090/S0002-9939-1977-0437696-1
- MathSciNet review: 0437696