Vol. 27, No. 1, 1968

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ISSN: 0030-8730
A theorem of Rolle’s type in En for functions of the class C1

G. P. Szegő

Vol. 27 (1968), No. 1, 193–195
Abstract

In a note recently published W. Leighton presents the following theorem.

Theorem 1. Let f(x) = (x1,,xn) be of class C2 in En. Suppose that f(x) has an isolated relative minimum at x = 0, and that f(0) = 0. If there is a point x = 0 where f(x) = 0, then f(x) must have at least one critical point, finite or infinite, in addition to that at the origin.

The proof employed by Prof. Leighton of Theorem (1) is based upon the theory of Morse and for this the condition that f(x) belongs to class C2 is essential.

In what follows we shall give a proof of Theorem (1) for functions of class C1.

Mathematical Subject Classification
Primary: 49.10
Secondary: 26.00
Milestones
Received: 10 May 1967
Revised: 24 November 1967
Published: 1 October 1968
Authors
G. P. Szegő