Volterra's Function

In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function ''V'' defined on the real line R with the following curious combination of properties:

* ''V'' is differentiable everywhere
* The derivative ''V'' ′ is bounded everywhere
* The derivative is not Riemann-integrable.

Definition and construction

The function is defined by making use of the Smith–Volterra–Cantor set and an infinite number or "copies" of sections of the function defined by

:$f(x) = \begin x^2 \sin(1/x), & x \ne 0 \\ 0, & x = 0.\end$

The construction of ''V'' begins by determining the largest value of ''x'' in the interval , 1/8for which ''f'' ′(''x'') = 0. Once this value (say ''x''₀) is determined, extend the function to the right with a constant value of ''f''(''x''₀) up to and including the point 1/8. Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0. This function will be defined to be 0 outside of the interval , 1/4 We then translate this function to the interval /8, 5/8so that the resulting function, which we call ''f''₁, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set.

To construct ''f''₂, ''f'' ′ is then considered on the smaller interval ,1/32 truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to ''f''₁ to produce the function ''f''₂. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function ''V'' is the limit of the sequence of functions ''f''₁, ''f''₂, ...

Further properties

Volterra's function is differentiable everywhere just as ''f'' (as defined above) is. One can show that ''f'' ′(''x'') = 2''x'' sin(1/''x'') - cos(1/''x'') for ''x'' ≠ 0, which means that in any neighborhood of zero, there are points where ''f'' ′ takes values 1 and −1. Thus there are points where ''V'' ′ takes values 1 and −1 in every neighborhood of each of the endpoints of intervals removed in the construction of the Smith–Volterra–Cantor set ''S''. In fact, ''V'' ′ is discontinuous at every point of ''S'', even though ''V'' itself is differentiable at every point of ''S'', with derivative 0. However, ''V'' ′ is continuous on each interval removed in the construction of ''S'', so the set of discontinuities of ''V'' ′ is equal to ''S''.

Since the Smith–Volterra–Cantor set ''S'' has positive Lebesgue measure, this means that ''V'' ′ is discontinuous on a set of positive measure. By Lebesgue's criterion for Riemann integrability, ''V'' ′ is not Riemann integrable. If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set ''C'' in place of the "fat" (positive-measure) Cantor set ''S'', one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set ''C'' instead of the positive-measure set ''S'', and so the resulting function would have a Riemann integrable derivative.

See also

* Fundamental theorem of calculus

References

External links

''Wrestling with the Fundamental Theorem of Calculus: Volterra's function''
, talk by David Marius Bressoud

''Volterra's example of a derivative that is not integrable''
{{Webarchive, url=https://web.archive.org/web/20160303185034/http://www.macalester.edu/~bressoud/talks/apnc2004/Volterra.ppt , date=2016-03-03 (PPT), talk by David Marius Bressoud

Fractals
Measure theory
General topology