From: Ronald Bruck Subject: Re: Existence of the directional derivatives Date: Sun, 06 Aug 2000 20:17:58 -0700 Newsgroups: sci.math Summary: [missing] In article <8ml4tj$4f3$1@nnrp1.deja.com>, bsguy@my-deja.com wrote: :I have a question and would like your help, please. : :Suppose f is a real valued function of several variables and a is an :element of its domain. In order for all directional derivatives to :exist at a, is it necessary that the partial derivatives exist and be :continuous in a neighborhood of a, or is it enough that the partial :derivatives just exist at a? Many Calculus and Real Analysis books are :somewhat vague on this point. I think the partial derivatives must be :continuous, but I'm not sure. Neither. It's not even NECESSARY that partial derivatives exist throughout a neighborhood of a. Consider, for example, (x^2 + y^2)(f(x)+f(y)), where f is continuous but nowhere differentiable. This function is differentiable at (0,0), but neither partial derivative exists at any other point. Nor is it enough that the partial derivatives just exist at a. To visualize this, consider the function min(|x|,|y|). As for the last part of your post (omitted), it's a matter of preference. You're speaking of the Fr\'echet derivative here, incidentally; the Gateau derivative is also of interest. --Ron Bruck -- Due to University fiscal constraints, .sigs may not be exceed one line.