Volterra's function: Difference between revisions

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{{Short description|Differentiable function whose derivative is not Riemann integrable}}

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

{{more citations needed|date=August 2024}}

[[File:Volterra function.svg|thumb|400px|right|The first three steps in the construction of Volterra's function.]]

* ''V''(''x'') is [[differentiable]] everywhere

* The derivative ''V'' ′(''x'') is [[bounded function|bounded]] everywhere

In [[mathematics]], '''Volterra's function''', named for [[Vito Volterra]], is a [[real-valued function]] ''V''(''x'') [[Function of a real variable|defined on the [[real line]] '''R''' with the following curious combination of properties:

* ''V''(''x'') is [[Differentiable function|differentiable]] everywhere

* The derivative ''V'' ′(''x'') is [[bounded function|bounded]] everywhere

* The derivative is not [[Riemann integration|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

The function is defined by making use of the [[Smith-Volterra-Cantor set]] and "copies" of the function defined by ''f''(''x'') = ''x''²sin(1/''x'') for ''x'' ≠ 0 and ''f''(''x'') = 0 for ''x'' = 0. The construction of ''V''(''x'') begins by determining the largest value of ''x'' in the interval [0, 1/8] for 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, which we call ''f''₁(''x''), will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [3/8, 5/8] so that the function is nonzero only on the middle interval as removed by the [[Smith-Volterra-Cantor set|SVC]]. To construct ''f''₂(''x''), ''f'' ′(''x'') is then considered on the smaller interval 1/16 and two translated copies of the resulting function are added to ''f''₁(''x''). Volterra's function then results by repeating this procedure for every interval removed in the construction of the [[Smith-Volterra-Cantor set|SVC]].

:<math>f(x) = \begin{cases} x^2 \sin(1/x), & x \ne 0 \\ 0, & x = 0.\end{cases}</math>

The function is defined by making use of the [[Smith-Volterra-Cantor set]] and "copies" of the function defined by ''f''(''x'') = ''x''²sin(1/''x'') for ''x'' ≠ 0 and ''f''(''x'') = 0 for ''x'' = 0. The construction of ''V''(''x'') begins by determining the largest value of ''x'' in the interval [0, 1/8] for which ''f'' &prime;(''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, which we call ''f''₁(''x''), will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [3/8, 5/8] so that the resulting function, iswhich nonzerowe only on the middle interval as removed by the [[Smith-Volterra-Cantor set|SVC]]. To constructcall ''f''₂₁(''x''), ''f'' &prime;(''x'') is thennonzero consideredonly on the smallermiddle interval 1/16 and two translated copies of the resulting function are added to ''f''₁(''x''). Volterra's function then results by repeating this procedure for every interval removed in the constructioncomplement of the [[Smith-Volterra-CantorSmith–Volterra–Cantor set|SVC]].

To construct ''f''₂, ''f'' &prime; is then considered on the smaller interval [0,1/32], 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'' (''x'')as (defined above) is. TheOne derivativecan show that ''Vf'' &prime;(''x'') is= discontinuous2''x'' atsin(1/''x'') the- endpointscos(1/''x'') offor every''x'' interval≠ removed0, inwhich themeans constructionthat ofin theany [[Smith-Volterra-Cantorneighborhood set|SVC]]of zero, butthere the function is differentiable at theseare points withwhere value''f'' 0.&prime; takes Furthermore,values in1 anyand neighbourhood of such&minus;1. a pointThus there are points where ''V'' &prime;(''x'') takes values 1 and &minus;1. in Itevery followsneighborhood thatof iteach isof notthe possible,endpoints forof everyintervals εremoved >in 0,the to find a partitionconstruction of the real[[Smith–Volterra–Cantor lineset]] such''S''. thatIn fact, |''V'' &prime;( is discontinuous at every point of ''xS''₂), &minus;even though ''V'' &prime;(''x''₁)|itself <is εdifferentiable onat every intervalpoint of [''xS''₁, with derivative 0. However, ''xV''₂] of&prime; is continuous on each interval removed in the partition.construction of Therefore''S'', so the derivativeset of discontinuities of ''V'' &prime;(''x'') is notequal Riemannto integrable''S''.

Since the Smith–Volterra–Cantor set ''S'' has positive [[Lebesgue measure]], this means that ''V'' &prime; is discontinuous on a set of positive measure. By [[Riemann integral#Integrability|Lebesgue's criterion for Riemann integrability]], ''V'' &prime; 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==

A real-valued function is Riemann integrable if and only if it is bounded and continuous almost-everywhere (''i.e.'' everywhere except a set of [[measure theory|measure]] 0). Since ''V'' &prime;(''x'') is bounded, it follows that it must be discontinuous on a set of positive measure, so in particular the derivative of ''V''(''x'') is discontinuous at uncountably many points.

{{Reflist}}

==External links==

* [http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf ''Wrestling with the Fundamental Theorem of Calculus: Volterra's function''] {{Webarchive|url=https://web.archive.org/web/20201123131325/http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf |date=2020-11-23 }}, talk by [[David Bressoud|David Marius Bressoud]]

* http://www.macalester.edu/~bressoud/talks/Volterra-4.pdf

* [http://www.macalester.edu/~bressoud/talks/apnc2004/Volterra.ppt ''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 Bressoud|David Marius Bressoud]]

[[Category:Fractals]]