Re: second partial derivatives commute (Clairaut's Thm.)

In article <1162175668.392176.46810@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
G Patel <gaya.patel@xxxxxxxxx> wrote:

To guarantee f_xy (a,b) = f_yx (a,b), do we require that
f_xy, f_yx be
continuous at (a,b) or a stronger requirement ( continuity in some
neighbourhood of (a,b) )?

In Adams, "Calculus, A Complete Course" (third edition), sec.
13.4, the requirement is that f_{xy} and f_{yx} are
continuous at (a,b), while f, f_x and f_y are continuous in
a neighbourhood of (a,b).

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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