Volterra's function: Difference between revisions

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

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

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

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

* ''V'' is [[Differentiable function|differentiable]] everywhere

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==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 <math>f(x) = x^2 \sin(1/x)</math> for <math>x \neq 0</math> and <math>f(0) = 0</math>. The construction of ''V'' 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 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, 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'' &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, ...

:<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 <math>f(x) = x^2 \sin(1/x)</math> for <math>x \neq 0</math> and <math>f(0) = 0</math>. The construction of ''V'' 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 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, 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'' &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, ...

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'' (as defined above) is. One can show that ''f'' &prime;(''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'' &prime; takes values 1 and &minus;1. Thus there are points where ''V'' &prime; takes values 1 and &minus;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'' &prime; is discontinuous at every point of ''S'', even though ''V'' itself is differentiable at every point of ''S'', with derivative 0. However, ''V'' &prime; is continuous on each interval removed in the construction of ''S'', so the set of discontinuities of ''V'' &prime; is equal to ''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_integralRiemann 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==

{{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/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]]