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When I studied math in my university Bauman Moscow State Techinical University we used notion of directional derivative as a quantity which can be evaluated even for function which has one side derivative but is not differetiable. [1], p.142

But looks that people from other cultures refer to directional derivative as two side derivative where h or t can come to zero from left or right:

http://mathworld.wolfram.com/DirectionalDerivative.html https://en.wikipedia.org/wiki/Directional_derivative

So in fact the concept that I used is only differ from the definition provided is that I and other Russian people consider t or h->+0.

For some maybe historical reasons - english speaking world refer to directional derivative as characterstic how function change in both v and -v direction.

If so how should I refer to my case to change of the function only in direction v without touch -v at all?

Refrences

[1]Book written by http://www.mathnet.ru/php/person.phtml?personid=23029&option_lang=eng V Канатников, Крищенко, и др. Дифференциальное исчисление функций многих переменных

Konstantin Burlachenko
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    I'm sure that your non-English text didn't use the term "directional derivative" but a term in the language of the text which you translated as "directional derivative". – celtschk Apr 19 '17 at 12:13
  • Non-english reference which I provided has term which I translated "directional derivative". But by direction it used something like a ray in function domain from in the points in which we consider, not a line. This is a difference. And very important difference if consider non smooth function – Konstantin Burlachenko Apr 19 '17 at 12:21
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    Looking around on Wikipedia, I think the concept of semi-differentiability seems to fit. – celtschk Apr 19 '17 at 12:21
  • Thanks) So I will use this word when I will talk with my English speaking friends. – Konstantin Burlachenko Apr 19 '17 at 12:25
  • I have at hand a book of convex analysis (Hiriart-Urruty, Lemaréchal, Convex Analysis and Minimization Algorithms 1). In IV.4.2, the (French-speaking, I guess) authors and the translators needed to make the distinction between the two things you say. In a way, the solution they adopted is using "directional derivative" for $$\lim\limits_{t\to 0^+} \frac{f(x+tv)-f(x)}{t}$$ and "partial derivative" for $$\lim\limits_{t\to 0} \frac{f(x+tv)-f(x)}{t}$$ –  Apr 19 '17 at 13:16
  • Though, if wikipedia says it, you might be better sticking to semi-differentiability. –  Apr 19 '17 at 13:21

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