Definition talk:Piecewise Continuously Differentiable Function

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Other definitions of piecewise continuously differentiable

(The definition of Piecewise Continuously Differentiable Function was changed yesterday, so this section was made with references to the definition of Piecewise Continuously Differentiable Function in the previous version, which was:

$(1): \quad$ $f$ is continuous
$(2): \quad$ there exists a finite subdivision $\left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, $x_0 = a$ and $x_n = b$, such that $f$ is continuously differentiable on $\left[{x_{i−1} \,.\,.\, x_i}\right]$, the derivatives at $x_{i−1}$ and $x_i$ understood as one-sided derivatives, for every $i \in \left\{{1, \ldots, n}\right\}$. Ivar Sand (talk) 08:23, 31 October 2013 (UTC))


1. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Magazines for other definitions of piecewise continuously differentiable and found:

- Agarwal and O’Regan[1]:

  • (1) replaced by: $f$ is piecewise continuous (according to Definition 11.1 in the book, which means that $f$ is not required to be defined at the points $x_i$).
  • (2) replaced by: $f'$ is piecewise continuous (according to Definition 11.1 in the book, which means that $f'$ is not required to be defined at the points $x_i$).
  • Term used: piecewise smooth. (I used the search function of maa.org and used the search term "piecewise continuous".)
Ivar Sand (talk) 09:37, 12 August 2013 (UTC)

- Kaplan[2]:

  • (1) replaced by: $f$ is piecewise continuous.
  • (2) replaced by: $f$ is continuously differentiable on ($x_{i−1}..x_i$) and $f′$ has one-sided limit(s) at every $x_i$.
  • Term used: piecewise smooth. (I used the search function of maa.org.)
Ivar Sand (talk) 09:37, 12 August 2013 (UTC), and 20 August 2013 (UTC)


2. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Wikis_and_Encyclopedias for other definitions of "piecewise continuously differentiable" and found none.


3. I have found these on the Internet (I have done only a limited search):

- In Methods of Mathematical Physics, Differential Equations by Richard Courant and D. Hilbert[3]:

  • (2) is replaced by: The derivative of $f$ is a piecewise continuous function. Ivar Sand (talk) 10:27, 24 July 2013 (UTC)


- In Complex Made Simple by David C. Ullrich[4]:

  • [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).
  • $f'$ has one-sided limit(s) at every $x_i$.

Here, $f$ is a complex-valued function. --Ivar Sand (talk) 09:17, 3 September 2013 (UTC)


- In Mathematics in Population Biology by Horst R. Thieme[5]:

  • [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).
  • Observation: $f'$ is allowed to exist but be discontinuous at some point $x_i$ where i∈{1,…,n-1}.


- In Analysis II by Herbert Amann and Joachim Escher[6]:

  • (1) is replaced by: $f$ is piecewise continuous,
  • $f$ is continuously differentiable on [$x_{i−1}..x_i$] in (2) replaced by $f'$ is uniformly continuous on ($x_{i−1}..x_i$).


- In A First Course in Harmonic Analysis by Anton Deitmar[7]:

  • (This seems not to be a different definition, only a reformulation). Ivar Sand (talk) 08:24, 26 July 2013 (UTC)


4. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Wikis_and_Encyclopedias for other definitions of "piecewise continuously differentiable" by searching for "piecewise smooth", which is sometimes synonymous with "piecewise continuously differentiable" and found:

- scholarpedia.org[8]:

  • [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).


- planetmath.org[9]:

  • (This seems not to be a different definition, only a reformulation).


5. I have found these on the Internet (I have done only a limited search):

- In Linear Partial Differential Equations for Scientists and Engineers (2007) by Tyn Myint-U and Lokenath Debnath: [10]:

  • (1) is replaced by: $f$ is piecewise continuous,
  • [$x_{i−1}..x_i$] in (2) is replaced by ($x_{i−1}..x_i$),
  • included in (2): the one-sided limits $f'(x_{i−1}+)$ and $f'(x_i-)$ exist for every $i \in \{1, \ldots, n\}$.

- Logg[11]:

  • (1) is replaced by: $f$ is piecewise continuous,
  • [$x_{i−1}..x_i$] in (2) is replaced by ($x_{i−1}..x_i$),
  • $f'$ is required to be bounded on the intervals ($x_{i−1}..x_i$).
Ivar Sand (talk) 10:59, 2 September 2013 (UTC)

Ivar Sand (talk) 10:27, 24 July 2013 (UTC)


Very nice and thorough work indeed. I guess the conclusion for our enterprise is that we need to be investigative as to the necessary assumptions for each theorem that uses this terminology. This page is to be expanded upon to indicate the non-universality of the terms -- particularly "piecewise smooth". — Lord_Farin (talk) 14:23, 12 August 2013 (UTC)
To be honest, the reason why I made the survey was that at the time when I registered the definition of piecewise continuously differentiable function I believed that there was only one such definition. I thought the least I could do was to make a survey of some of the other definitions of piecewise continuously differentiable function and put the survey on the talk page. Ivar Sand (talk) 07:47, 14 August 2013 (UTC)
The definition currently up is the most natural one to me as well, but perhaps in the future we will see the need to distinguish between, say, continuous, piecewise continuously differentiable function and piecewise continuous, piecewise continuously differentiable function (and both these names are craving for acronyms, e.g. cPCD and pcPCD). — Lord_Farin (talk) 08:08, 14 August 2013 (UTC)
I have taken a close look at the category of definitions of Piecewise Continuously Differentiable Function above that require $f$ to be continuous. I call this category the continuity definition category. I use my own notes above and hope that they are correct. I found:
  • The reference numbers of the definitions above that belong to the continuity definition category, are [3], [4], [5], [7], [8], and [9].
  • [7] and [9] are equivalent to the definition of the definition page.
  • The difference between [3] and [4] is confined to (2) in the definition in the definition page. Restricted to this part of the definition, [3] says that $f'$ is continuous on the intervals ($x_{i−1}..x_i$) and that the one-sided limits $f′(x_{i−1}+)$ and $f′(x_i−)$ exist. Correspondingly, [4] says that $f'$ is continuous on ($x_{i−1}..x_i$) and that $f'$ has one-sided limit(s) at every $x_i$. Accordingly, [3] and [4] say the same thing and are equivalent.
  • The difference between [3]/[4] and [7]/[9] is that [3]/[4] lacks the requirement of [7]/[9] that the one-sided derivatives of $f$ at the points $x_i$ exist. However, this requirement is unnecessary as it is a proven fact and follows from the Extension of Derivative theorem[12]. This theorem, or rather a version of it that fits our purposes, says that if a function $f$ is continuous at a point x and the limit of $f'$ from one side, say the right, exists, then the right-derivative of $f$ at x exists as well and equals this limit. Therefore, [3]/[4] is equivalent to [7]/[9] (and the definition of the definition page could be changed into [3] or [4], which is less restrictive).
  • The definitions [5] and [8] are equivalent. Note that the [5]/[8] definition places no restrictions on the one-sided derivatives of $f$ at the points $x_i$ and no restrictions on the one-sided limits of $f'(x)$ as x approaches $x_i$. In particular, the [5]/[8] definition allows $f'$ to be unbounded.
  • In conclusion, the continuity definition category consists of 2 definitions: [3]/[4]/[7]/[9] and [5]/[8]. Ivar Sand (talk) 08:58, 15 August 2013 (UTC)


I have taken a close look at the definitions of Piecewise Continuously Differentiable Function above that allow $f$ to be piecewise continuous. I call this category of definitions the piecewise continuous category. I use my own notes above and hope that they are correct.
  • The reference numbers of the definitions that belong to the piecewise continuous category, are [1], [2], [6], [10], and [11].
  • [2] looks like this:
(1): $f$ is piecewise continuous
(2): $f$ is continuously differentiable on ($x_{i−1}..x_i$) and the one-sided limits $f'(x_{i−1}+)$ and $f'(x_i−)$ exist.
  • A close inspection of [1] reveals that [1] differs from [2] only in that it allows $f$ to be undefined at the points $x_i$.
  • [10] is equivalent to [2].
  • [6] is more complicated, but the following detailed inspection leads to the conclusion that [6] too is equivalent to [2]. The uniform continuity of $f′$ on the open intervals ($x_{i−1}..x_i$) as is required by [6] implies by using Cauchy sequences that the one-sided limits $f′(x_{i−1}+)$ and $f′(x_i−)$ exist.[13] Therefore, [6] implies [2]. Moreover, starting from (2) in [2], since the one-sided limits $f′(x_{i−1}+)$ and $f′(x_i−)$ exist $f′$ can be extended to a function $f^*$ that satisfies: $f^*$ equals $f′$ on ($x_{i−1}..x_i$), $f^*(x_{i−1})$ equals $f′(x_{i−1}+)$, and $f^*(x_i)$ equals $f′(x_i−)$. $f^*$ is continuous on [$x_{i−1}..x_i$]. Since a continuous function defined on a closed interval is uniformly continuous,[14] $f^*$ is uniformly continuous on [$x_{i−1}..x_i$]. This implies that $f′$ is uniformly continuous on ($x_{i−1}..x_i$). Therefore, [6] adds nothing new and is equivalent to [2].
  • In conclusion, the piecewise continuous category consists of the three definitions [2]/[6]/[10], [1], and [11]. [1] differs from [2]/[6]/[10] in that it allows $f$ to be undefined at the points $x_i$. [11] differs from [2]/[6]/[10] in that it replaces the one-sided limit requirements in [2]/[6]/[10] with the requirement that $f'$ be bounded. Ivar Sand (talk) 09:55, 20 August 2013 (UTC) and 2 September 2013 (UTC)
  • Conclusion:
  1. The piecewise continuous category consists of the three definitions [2]/[6]/[10], [1], and [11].
  2. [1] differs from [2]/[6]/[10] in that it allows $f$ to be undefined at the points $x_i$.
  3. [11] differs from [2]/[6]/[10] in that the one-sided limit requirements in [2]/[6]/[10] are replaced by the requirement that $f'$ be bounded.
Ivar Sand (talk) 09:55, 20 August 2013 (UTC) and 2 September 2013 (UTC)

Multiple definitions

I feel the same approach should be taken here as for Definition:Piecewise Continuous Function. That is, the respective equivalent definitions need to be identified and given appropriate, distinct names. If that turns out to be very hard or impossible, we will have to invent a new approach. But only as and when that becomes necessary. It's probably best to wait until the PC function definition has crystallised, so that we can take on board any leassons learnt there. — Lord_Farin (talk) 16:19, 26 May 2015 (UTC)

References

  1. Ravi P. Agarwal, Donal O’Regan: Ordinary and Partial Differential Equations (2009): §11: Definition 11.2
  2. Wilfred Kaplan: Advanced Calculus (Fifth Edition): §7.3
  3. http://books.google.no/books?id=fcZV4ohrerwC&pg=PA245&dq=piecewise+Richard+Courant&hl=no&sa=X&ei=k3HuUZGJCIqp4ASrlYHACw&ved=0CDYQ6AEwAQ#v=onepage&q=piecewise%20Richard%20Courant&f=false
  4. http://books.google.no/books?id=-eNiTAAioRUC&pg=PA17&dq=%22piecewise+continuously+differentiable%22&hl=no&sa=X&ei=NwDpUfvTBIqi4gSu04D4BA&ved=0CHEQ6AEwCA#v=onepage&q=%22piecewise%20continuously%20differentiable%22&f=false
  5. http://books.google.no/books?id=cHcjnkrMweYC&pg=PA491&dq=%22piecewise+continuously+differentiable%22&hl=no&sa=X&ei=NwDpUfvTBIqi4gSu04D4BA&ved=0CGIQ6AEwBg#v=onepage&q=%22piecewise%20continuously%20differentiable%22&f=false
  6. http://books.google.no/books?id=mCmEZ-UtFBEC&pg=PA83&dq=%22piecewise+continuously+differentiable%22&hl=no&sa=X&ei=NwDpUfvTBIqi4gSu04D4BA&ved=0CEcQ6AEwAw#v=onepage&q=%22piecewise%20continuously%20differentiable%22&f=false
  7. http://books.google.no/books?id=zw_wjr_k9BYC&pg=PA17&dq=%22piecewise+continuously+differentiable%22&hl=no&sa=X&ei=NwDpUfvTBIqi4gSu04D4BA&ved=0CGkQ6AEwBw#v=onepage&q=%22piecewise%20continuously%20differentiable%22&f=false
  8. http://www.scholarpedia.org/article/Piecewise_smooth_systems
  9. http://planetmath.org/piecewisesmooth
  10. http://link.springer.com/search?query=piecewise+smooth&search-within=Book&facet-publication-title=%22Linear+Partial+Differential+Equations+for+Scientists+and+Engineers%22#page-1
  11. http://home.simula.no/~logg/pub/thesis/compact-disc/papers/pdf/paper-07.pdf
  12. http://clem.mscd.edu/~talmanl/TeachCalculus/Chapter040.pdf
  13. http://math.stackexchange.com/questions/178266/extending-continuous-and-uniformly-continuous-functions
  14. 1967: Tom M. Apostol: Calculus Volume 1: $\S 3.17$: Theorem $3.13$

Cut the Gordian knot

I lack the personal qualities that allow me to spend the necessary time studying the above in detail.

However, I have looked at the various sources I have, and the following observations can be made.

There are two main motivations at the elementary level for Definition:Piecewise Continuously Differentiable Function.

a) Remmert's definition is in the context of paths in the complex plane. It demands that the path itself is continuous to start with. Hence the requirement that $f$ itself be continuous is a specialisation of the more general case where $f$ need not be continuous, but in fact only piecewise continuous (with one-sided limits at the end points)

Consequently, his definition leads to the fact that $\displaystyle \lim_{x \mathop \to x_r^+} f \left({x}\right)$ on $\left({x_{r - 1} \,.\,.\, x_r}\right)$ equals $\displaystyle \lim_{x \mathop \to x_r^-} f \left({x}\right)$ on $\left({x_r \,.\,.\, x_{r + 1}}\right)$ because $f \left({x_r}\right)$ is defined a fortiori.

b) In the context of Fourier analysis, the game is different. There is no requirement that $f \left({x}\right)$ is continuous in order for Fourier analysis to "work". In fact, $f \left({x_r}\right)$ may not even be defined at all -- as long as $f$ is defined on both $\left({x_{r - 1} \,.\,.\, x_r}\right)$ and $\left({x_r \,.\,.\, x_{r + 1}}\right)$, and that the one-sided limits exist.

Consequently, for $f$ to be a piecewise continuously differentiable function, it is only required that, for all $r$, $\left({x_{r - 1} \,.\,.\, x_r}\right)$ has right-hand derivative at $x_r$, and a left-hand derivative at $x_{r - 1}$, but this is possible even if $f \left({x_r}\right)$ and $f \left({x_{r - 1} }\right)$ do not exist. This is the whole point of Fourier analysis.

For Fourier's Theorem, if $f$ is piecewise continuous, and is continuously differentiable on all internal points of the (open) intervals that constitute the subdivision, and has right-hand derivatives and left-hand derivatives at the endpoints at each interval, then the Fourier series which approximates $f$ provides a well-defined value for $f$ at the endpoints of those intervals: the arithmetic mean of the one-sided limits at the endpoint of the interval either side of the discontinuity.

But while it is necessary that those limits exist (hence $f$ is needed to be piecewise continuous with one-sided limits), it is not necessary for $f$ to actually be defined at those endpoints.


In conclusion ...

I believe there is only one definition needed here for $f$ to be a piecewise continuously differentiable function:

$f$ is piecewise continuous (in the weakest sense -- one-sided limits follow from the other conditions a fortiori)
$f$ is continuously differentiable everywhere on the (open) intervals
$f$ has right-hand derivatives and left-hand derivatives at the endpoints at each interval.

And, to recapitulate, the fact that $f$ has a right-hand derivative or left-hand derivative at $x_r$ (or even both) does not mean $f$ is actually continuous at $x_r$ or even defined on $x_r$.


If we do need the definition to include continuity of $f$, for example in the context of complex differentiation over a contour $f \left[{I}\right]$, then we can define $f$ to be:

a) Continuous on $I$
b) Piecewise continuously differentiable on $I$.

I do not believe it is necessary to make a completely separate definition for "continuous and piecewise continously differentiable function" any more than it is necessary to introduce a new definition for "odd positive number", for example. A number may be both odd and positive, and if the requirement of the proof is such as to require that this is the case, we would start our proof by saying "let $n$ be an odd integer such that $n$ is positive" or the other way around.

However, it may be convenient to define a "continuous and piecewise continously differentiable function", because these have a specific use in complex analysis. Just that such a definition is not in itself a definition of a "piecewise continously differentiable function", merely a specialisation of it.


Apologies for having gone on so long on this subject -- I was thinking it through as I was writing it, and clarifying my own thoughts on the matter as I went.

Please feel free to comment, therefore, on my suggestion that a piecewise continuously differentiable function be defined as I have suggested above, and that the definition as provided on this page be introduced as a composite definition of "continuous and piecewise continously differentiable". --prime mover (talk) 04:35, 22 March 2018 (EDT)

Note that if $f$ has a left-hand derivative or a right-hand derivative at a point, $f$ is defined at that point according to the definitions of left-hand derivative and right-hand derivative.
Also, if $f$ has a left-hand derivative and a right-hand derivative at a point, $f$ is continuous at that point by Left-Hand and Right-Hand Differentiable Function is Continuous. --Ivar Sand (talk) 04:52, 28 March 2018 (EDT)
Yes. And? --prime mover (talk) 09:58, 28 March 2018 (EDT)
What this means for instance for your suggested definition of piecewise continuously differentiable function is that its first ($f$ is piecewise continuous ...) and third ($f$ has right-hand derivatives and ...) requirements are inconsistent.
It seems to me that you either have to remove "piecewise" from the first requirement or replace "and" with "exclusive or" in the third requirement. --Ivar Sand (talk) 17:53, 28 March 2018 (EDT)
Okay, someone else do fourier analysis. I've had enough. --prime mover (talk) 00:56, 29 March 2018 (EDT)

Actually, the whole analysis you did seems sound. The only problem is the currently provided definition of left- and right-hand derivative. However I do think that it will be necessary to have one-sided limits to be able to repair the definition satisfactorily (which makes sense from the perspective that differentiation is above continuity).

This is not in contradiction with the cited continuity theorems because they rely on $a \in I$ being an internal point of some open interval. More critically, the failure of that approach depends on that $f(a)$ in said cases cannot be meaningfully defined, even though it can be from either one-sided perspective.

Given these points, I think PM's proposal should be reconsidered, imagining a "fixed"/"generalised" definition of one-sided derivative. — Lord_Farin (talk) 13:58, 8 April 2018 (EDT)