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There is a well-known example of continuous (Weierstrass) function having no derivative at any point.There is also a continuous function having no left nor right derivative at any point,but I cannot find the construction.Where can I find it and how such a pathological curve is called?

Amzoti
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user122424
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  • I think the first answer to this question gives such a construction. – Clement C. Mar 19 '15 at 12:21
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    But by $(c)$ there:$$\text{(c)} ;;;; f'(x) = 0 ;;; \text {for almost all} ; x \in [0,1],$$so this is not an example of the problem,and $(d),(e)$ do not give answer either.Which particular part of first answer is relevant? – user122424 Mar 19 '15 at 12:41
  • Look further down, the (d) and (e) parts (Admittedly, it's not at every point, but only on dense subsets.): "Second, (d) implies that f has no left derivative (finite or infinite) at densely many points in [0,1] and (e) implies that f has no right derivative (finite or infinite) at densely many points." – Clement C. Mar 19 '15 at 12:58
  • But is there a construction of our $f$ given in $(d)$ and $(e)$ or it just says non-constructively that all but a first Baire category set of functions in the space of non-decreasing continuous functions have this property? – user122424 Mar 19 '15 at 13:11
  • As far as I can see, it is non-constructive (I searched for an explicit construction, with no luck so far). – Clement C. Mar 19 '15 at 13:13
  • If we knew the name for our $f$,life would be easier.I think that it is not a very complicated construction,and it holds everywhere in $(0,1)$.It certainly involves Dini's derivations. – user122424 Mar 19 '15 at 13:15
  • see my answer below -- an explicit construction is given in the linked paper. – Clement C. Mar 19 '15 at 13:37

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From Derivatives (A. M. Bruckner and J. L. Leonard, 1966), on p.39:

Even this last remnant of good behavior can be removed, as in Besicovitch's example [6,147] which at no point has even a unilateral derivative (even infinite).

[6] is in Russian, but [147] refers to E. Pepper's On continuous functions without a derivative, which gives another proof of Besicovitch's result. It looks like (?) these functions are often referred to as Besicovitch functions.

See also Chapter 13, p.144 of Differentiation of Real Functions (Andrew M. Bruckner, 1978) for some more references and a short discussion.

Clement C.
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