Definition talk:Piecewise Continuously Differentiable Function

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) 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 :
 * (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 :


 * (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 :

Here, $f$ is a complex-valued function. --Ivar Sand (talk) 09:17, 3 September 2013 (UTC)
 * [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).
 * $f'$ has one-sided limit(s) at every $x_i$.

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


 * [$x_{i−1}..x_i$] 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 :


 * (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 :


 * (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 :


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

- planetmath.org :


 * (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: :


 * (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 :
 * (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". &mdash; 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). &mdash; 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 . 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. 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, $f^*$ 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)