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 $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that $f$ is continuously differentiable on $\closedint {x_{i - 1} } {x_i}$, the derivatives at $x_{i - 1}$ and $x_i$ understood as one-sided derivatives, for every $i \in \set {1, \ldots, n}$. Ivar Sand (talk) 08:23, 31 October 2013 (UTC)

This definition is essentially the same as the current Definition 2.)

1. I have searched the list at Book:Other Websites#Magazines for other definitions of piecewise continuously differentiable and found:

- Agarwal and O’Regan:

• (1) replaced by: $f$ is piecewise continuous (with one-sided limits) (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 (with one-sided limits) (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 (with one-sided limits).
• (2) replaced by: $f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$ 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 Book:Other Websites/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 (the general definition). Ivar Sand (talk) 10:27, 24 July 2013 (UTC)

- In Complex Made Simple by David C. Ullrich:

• $\closedint {x_{i - 1} } {x_i}$ in (2) replaced by $\openint {x_{i - 1} } {x_i}$.
• $f'$ 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:

• $\closedint {x_{i - 1} } {x_i}$ in (2) replaced by $\openint {x_{i - 1} } {x_i}$.
• Observation: $f'$ is allowed to exist but be discontinuous at some point $x_i$ where $i \in \set {1, \ldots, n - 1}$.

- In Analysis II by Herbert Amann and Joachim Escher:

• (1) is replaced by: $f$ is piecewise continuous (the general definition),
• $f$ is continuously differentiable on $\closedint {x_{i - 1} } {x_i}$ in (2) replaced by $f'$ is uniformly continuous on $\openint {x_{i - 1} } {x_i}$.

Here, $f$ is a complex-valued function.

- 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 Book:Other Websites/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:

• (1) is replaced by: $f$ is piecewise continuous (with one-sided limits),

• $\closedint {x_{i - 1} } {x_i}$ in (2) replaced by $\openint {x_{i - 1} } {x_i}$.

An inspection of Figure 1 (b) in reference [8] reveals that $f$ may be piecewise continuous. --Ivar Sand (talk) 09:29, 17 November 2020 (UTC)

The identification of the type of piecewise continuity as the one with one-sided limits is not taken from a formal definition but from the figures in Figure 1, which shows examples of piecewise-smooth one-dimensional maps.

- 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 (with one-sided limits),

• $\closedint {x_{i - 1} } {x_i}$ in (2) is replaced by $\openint {x_{i - 1} } {x_i}$,

• 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 (bounded),
• $\closedint {x_{i - 1} } {x_i}$ in (2) is replaced by $\openint {x_{i - 1} } {x_i}$,
• the one-sided limits $f'(x_{i - 1}+)$ and $f'(x_i-)$ exist for every $i \in \{1, \ldots, n\}$
• the one-sided limits $f'(x_i+)$ and $f'(x_i-)$ are equal for every $i \in \{1, \ldots, n - 1\}$.

$f'$ is called continuous in reference [11] obviously because it is easily extendable to a continuous function.

Ivar Sand (talk) 10:59, 2 September 2013 (UTC), --Ivar Sand (talk) 10:29, 16 November 2020 (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], and [9].
• [7] and [9] are equal to the definition of the definition page so they are not new definitions and therefore of no interest here.
• 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 $\openint {x_{i - 1} } {x_i}$. Correspondingly, [4] says that $f'$ is continuous on $\openint {x_{i - 1} } {x_i}$ and that $f'$ has one-sided limit(s) at every $x_i$.
• [4] lacks the requirement of the definition of the definition page 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, [4] is equivalent to the definition of the definition page.
• [3] and [5] are equal.
• [4] is equal to Definition 1 so we already have that one.
• In conclusion, the continuity definition category consists of only one definition: [3]/[5], which is:

(1): $f$ is continuous

(2): $f$ is continuously differentiable on every open interval $\openint {x_{i - 1} } {x_i}$. Ivar Sand (talk) 08:58, 15 August 2013 (UTC), --Ivar Sand (talk) 10:56, 6 November 2020 (UTC), --Ivar Sand (talk) 11:41, 18 November 2020 (UTC), --Ivar Sand (talk) 10:23, 3 December 2020 (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], [8], [10], and [11].
• [2] looks like this:

(1): $f$ is piecewise continuous (with one-sided limits)

(2): $f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$ 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 $x_i$. As for Definition:Piecewise Continuous Function, I take $f$ being undefined as a variation as stated below. So I consider [1] and [2] to be equal.
• [10] is equal to [2].
• [6] and [2] are similar, as is revealed by a detailed inspection not given here. However, they are different as [6] requires $f$ to be piecewise continuous according to the general definition whereas [2] requires $f$ to be piecewise continuous with one-sided limits.
• [1]/[2]/[10] differs from [8] in that it requires that the one-sided limits $f'(x_i+)$ and $f'(x_i-)$ exist.
• [11] differs from [1]/[2]/[10] in e.g. that it requires the one-sided limits $f'(x_i+)$ and $f'(x_i-)$ to be equal. (Maybe piecewise continuous (bounded) in the definition of reference [11] could be changed to piecewise continuous (the general definition) by introducing a theorem that proves the boundedness.)
• In conclusion, the piecewise continuous category consists of the four definitions [1]/[2]/[10], [6], [8] and [11].

These definitions are:

[6]:

(1): $f$ is piecewise continuous (the general definition)

(2): $f'$ is uniformly continuous on $\openint {x_{i - 1} } {x_i}$.

[11]:

(1): $f$ is piecewise continuous (bounded)

(2): $f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$,

(3): the one-sided limits $f'(x_{i - 1}+)$ and $f'(x_i-)$ exist,

(4): the one-sided limits $f'(x_i+)$ and $f'(x_i-)$ are equal.

[8]:

(1): $f$ is piecewise continuous (with one-sided limits)

(2): $f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$.

[1]/[2]/[10]:

(1): $f$ is piecewise continuous (with one-sided limits)

(2): $f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$,

(3): the one-sided limits $f'(x_{i - 1}+)$ and $f'(x_i-)$ exist. Ivar Sand (talk) 09:55, 20 August 2013 (UTC) and 2 September 2013 (UTC), --Ivar Sand (talk) 10:29, 16 November 2020 (UTC), --Ivar Sand (talk) 10:23, 3 December 2020 (UTC)

Ivar Sand (talk) 09:55, 20 August 2013 (UTC) and 2 September 2013 (UTC)

Below it is suggested that the definitions of Piecewise Continuously Differentiable Function should not have requirements of continuity or piecewise continuity. This would have the following implications:

• For existing definitions:

The continuity requirement (Requirement (1)) of Definition 1 is removed. Then a comment should be included for the reference in the Source section of Definition 1, stating that the definition of Piecewise Continuously Differentiable Function given there includes the additional requirement of continuity.

• For the definitions of the continuity definition category:

Definition [3]/[5] should be changed in the same way as Definition 1. That way, the definitions of the continuity definition category would not talk about continuity any more but the definitions they refer to would.

• For the definitions of the piecewise continuous category:

The piecewise continuous category should be changed in the same way as the continuity definition category. That is, that the definitions of the piecewise continuous category should not talk about piecewise continuity any more but the definitions they refer to would. Anyway, every definition of the piecewise continuous category imply piecewise continuity. A theorem that proves it could be included and it would be referred to for every definition of the piecewise continuous category.

This would affect the definitions of the piecewise continuous category like this:

- All: (1) is removed

- [8] equals [3]/[5] and is merged with [3]/[5] to produce [3]/[5]/[8]

- [1]/[2]/[10] equals Definition 1, and references [1], [2] and [10] are included in the Source section of Definition 1.

Thus the number of definitions in the piecewise continuous category is reduced from 4 to 2.

With the one definition in the continuity definition category, it makes 3 new definitions in all. Therefore, there would be 5 definitions in all on the definition page for Piecewise Continuously Differentiable Function. --Ivar Sand (talk) 11:15, 13 January 2022 (UTC)

This is the updated list of definitions for piecewise continuously differentiable function:

• [3]/[5]/[8]:

$f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$

• Definition 1/[1]/[2]/[10]:

$f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$

the one-sided limits $f'(x_{i - 1}+)$ and $f'(x_i-)$ exist

• Definition 2:

$f$ is continuously differentiable on $\closedint {x_{i - 1} } {x_i}$

• [11]:

$f$ is continuously differentiable on $\openint {x_{i - 1} } {x_i}$

the one-sided limits $f'(x_{i - 1}+)$ and $f'(x_i-)$ exist

the one-sided limits $f'(x_i+)$ and $f'(x_i-)$ are equal

• [6]:

$f'$ is uniformly continuous on $\openint {x_{i - 1} } {x_i}$

One definition in the list implies the next one above it starting at Definition 2.

[11] implies Definition 1/[1]/[2]/[10].

I think that it might be possible to prove that [6] is equivalent to Definition 1/[1]/[2]/[10].

I suggest the following new names for the definitions for piecewise continuously differentiable function:

• [3]/[5]/[8]:

Piecewise Continuously Differentiable Function

• Definition 1/[1]/[2]/[10]:

Piecewise Continuously Differentiable Function with One-Sided Limits

• Definition 2:

Piecewise Continuously Differentiable Function on Closed Intervals

• [11]:

Piecewise Continuously Differentiable Function with Equal One-Sided Limits

• [6]:

Piecewise Continuously Differentiable Function with Uniform Continuity --Ivar Sand (talk) 09:21, 20 January 2022 (UTC)

Variations:

• $f$ is a complex-valued function in reference [4] in the continuity definition category.
• $f$ is a complex-valued function in reference [6] in the piecewise continuous category.
• $f$ is allowed to be undefined at the points $x_i$ in reference [1] in the piecewise continuous category.

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)

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 $\ds \lim_{x \mathop \to x_r^+} \map f x$ on $\openint {x_{r - 1} } {x_r}$ equals $\ds \lim_{x \mathop \to x_r^-} \map f x$ on $\openint {x_r} {x_{r + 1} }$ because $\map f {x_r}$ is defined a fortiori.

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

Consequently, for $f$ to be a piecewise continuously differentiable function, it is only required that, for all $r$, $\openint {x_{r - 1} } {x_r}$ has right-hand derivative at $x_r$, and a left-hand derivative at $x_{r - 1}$, but this is possible even if $\map f {x_r}$ and $\map f {x_{r - 1} }$ 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 $f \sqbrk I$, 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)

Then I think a "fixed"/"generalised" definition of one-sided derivative should be defined on its own page so that it can be discussed. The page should make clear when the concept should be used and when it should not be used. I admit that I don't understand any of this myself.--Ivar Sand (talk) 10:22, 12 January 2022 (UTC)

I agree that continuity should not be part of the definition of piecewise continuously differentiable function. One reason for this is that continuity is not a piecewise property (see piecewise). Another reason for this, which I believe is your idea, is that continuity is not a property of differentiability but a separate property.

I think the requirement for piecewise continuity should be removed too from the definition of piecewise continuously differentiable function because piecewise continuity is not a property of piecewise differentiability but a separate property. Instead, a theorem should be included that states that a piecewise continuously differentiable function is piecewise continuous. This theorem should then be referenced for each definition of piecewise continuously differentiable function. --Ivar Sand (talk) 10:47, 12 January 2022 (UTC)

You say that your definition should be the only one for piecewise continuously differentiable function. This means that your definition would replace Definition 1 and Definition 2, right? --Ivar Sand (talk) 11:01, 14 January 2022 (UTC)

Since we (collectively) admit we don't know what we're doing, the prudent approach (and the one which we are going to take) is for us to use the definitions as they appear in the source works that we quote. We have two of these. Then we get rid of all the overcomplication, and move on. --prime mover (talk) 12:10, 14 January 2022 (UTC)

OK, but I'm not the right person to start the work on this new approach. --Ivar Sand (talk) 09:05, 20 January 2022 (UTC)

Do you not have access to either of those source works? Sorry, I thought you were the author of these pages in the first place. Not to worry then, leave it and someone else will do it. --prime mover (talk) 09:17, 20 January 2022 (UTC)

No, I don't have access to either of those source works. Yes, I was the author of these pages in the first place. --Ivar Sand (talk) 09:37, 21 January 2022 (UTC)