# Follow up on Fréchet derivative

I don't know whether my doubt warrants a separate question thread as it shares similarities with my last question. So in that case I apologise.

Fréchet derivative

The thing is as Siminore said a Fréchet derivative in general for infinite dimensional spaces is defined as a "Continuous linear map". Note that in this case derivative of a linear map is the map itself. So if one was asked whether its possible for a linear map to be Fréchet differentiable without being continuous, answer would be no according to me. However I came upon this on the web:

http://iopscience.iop.org/1064-5632/59/5/A11

I wasn't able to access the full text and anyway very little of the title made sense to me, but the heading seemed to indicate that I was wrong, for it speaks of an everywhere discontinuous function with Fréchet derivatives. Is this possible even if my function was linear? Also how would my question make sense in case of a Gâteaux derivative which doesn't assume linearity in the first place and lastly what is the necessity for defining a Gâteaux derivative?

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## 1 Answer

I can download the cited paper. There is no mystery at all: the function is defined on the Schwartz space of smooth functions with compact support. In other words, the author deals with differential calculus on locally convex topological vector spaces. It is a very hard subject, and it may be rather different than differential calculus on normed vector spaces. In addition, the terminology seems to be a bit peculiar, since Fréchet differentiability means something slightly different than as usual.

In a normed vector space, Fréchet differentiability implies continuity, while Gâteaux differentiability (in the sense of existence of all directional derivatives) need not imply any continuity.

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Thanks, your answer is more than satisfactory. I dont think you need go through the trouble of downloading the text, since for a beginner like me it wouldnt be of any consequence at this point of time. However I was interested in it from the point of view of the problem I had. – Vishesh Aug 31 '12 at 14:52