Talk:Piecewise

I found this article to be informative -- but very confusing.

I am trying to remove the confusion (lack of clarity) without introducing errors. Please help.

Why I changed what I changed:

We are talking about the definition of piecewise, I believe, not the definition of a piecewise function f(x), so we should make that clear.

If a word describes a property, it describes a noun, so that word must be an adjective, not an adverb. But the first meaning of piecewise we discuss is also an adjective, so it's confusing to say that the second meaning is an adjective.

We haven't defined interval, so we should avoid that word, but we've sort of defined piece, so it's probably okay to use that.

TH 20:20, 15 October 2006 (UTC)

Removed original 2nd paragraph for clarity; added to 1st para

Here's what the original sectond paragraph said:

"According to the standard definitions, this is a single function, that happens to have its value computed by different methods in different cases. It is useful to do this, for example to make a sawtooth function. That is an example of a piecewise linear function: its graph is made up of a number of parts of the graphs of linear functions. Problems can arise at the ends of the intervals used for separate definitions. We must give a definite value for f(x) there, as everywhere else. It may be a point where continuity fails (as for the Heaviside function at 0), or where the function isn't smooth (the absolute value function at 0)."

But we never say what the "standard definitions" are, nor the non-standard definitions (if such exist), so let's not refer to them, whatever they are. Bringing in the example of a sawtooth function adds no information beyond what we provided in the Heaviside example -- also a piecewise linear function.

Every function must be defined across its entire domain, so we add nothing but confusion by stating that a piecewise function must be defined across its entire domain.

We shouldn't talk about vague "problems arising" when those problems are well known and are exactly what we are tackling when we define piecewise functions. Smoothness (differentiability) is not an issue in the definition of a piecewise function, so it will confuse the reader if we bring it up (unless we want to say that -- oh so obvious from the examples -- a piecewise function need not be differentiable across its entire domain).

Is there a piecewise function whose major structure is not if-then-else? If so, please fix the article and provide (there or here in Discussion) an example. Otherwise, this is a critical fact, since many people think of functions as having only arithmetic-like (not logic-like) definitions.

TH 20:51, 15 October 2006 (UTC)

Simplified and clarified last paragraph

Here's what the last paragraph originally said:

"The definitions of piecewise continuous, piecewise differentiable and so on are therefore made, to require that the 'pieces' of the function are continuous (resp. differentiable), but that at the end points failure of those conditions is allowed. A path said to be piecewise continuously differentiable is a continuous path (in the plane, say) but which can at some points turn direction sharply, so the continuity of the derivative vector at those points doesn't hold".

Pretty clearly the original was implying in a number of places that piecewise and piecewise continuous are synonyms, so I made the synonymy explicit. The word "path" adds nothing to the reader's understanding of what we've been saying about a function. The phrase "in the plane, say" adds nothing to the reader's understanding, so I took it out. The word "sharply" is not very clear -- if we meant instantaneously, then that's just another (unnecessary) repetition of what we've been saying throughout the article.

TH 21:36, 15 October 2006 (UTC)

Definition

Isn't the following definition better:

A function f(x) is said to be piecewise P (with P = continuous, differentiable, and so forth) if the subset of the domain where it fails to be P only contains isolated points.

This would be my intuitive definition. --14:37, 24 August 2008 (UTC) —Preceding unsigned comment added by 80.101.100.53 (talk • contribs)

Piecewise Smooth?

I was redirected here from 'piecewise smooth.' I get what piecewise means now but what does piecewise smooth mean? —Preceding unsigned comment added by 216.204.189.42 (talk) 01:03, 6 November 2009 (UTC)

Inverse definition

If possible, could the technical term for a "non-piecewise function" be added? -- Robbiemorrison (talk) 10:45, 25 January 2011 (UTC)

piecewise

preciseness 49.145.160.177 (talk) 14:57, 6 September 2022 (UTC)

Article title

Discussion: Wikipedia_talk:WikiProject_Mathematics/Archive/2024/Mar#Piecewise. fgnievinski (talk) 03:43, 30 March 2024 (UTC)

Requested move 20 July 2024

Piecewise → Piecewise mathematical object – Per WP:NOUN. The adjective title is jarring. Previous discussion is at Wikipedia_talk:WikiProject_Mathematics/Archive/2024/Mar#Piecewise. Mathematical object should include everything, addressing concerns about the concept needing to cover piecewise linear manifold and piecewise linear curve 174.92.25.207 (talk) 14:56, 20 July 2024 (UTC)

• Oppose. This suggested title is much more jarring for a mathematician than the existing title. Moreover, "piecewise" qualifies generally a property, not a mathematical object. Grammatically, it is generally not used as an adjective, but as an adverb that qualifies an adjective ("piecewise function" seems an exception, but it is only an abbrevation of "piecewise-defined function")
• Suggestion. A move Piecewise → Piecewise property would satisfy WP:NOUN and better correspond to the mathematical meaning.

D.Lazard (talk) 15:42, 20 July 2024 (UTC)

• Turning into a disambig page. I don’t think there is a mathematical notion called "piecewise". What we have instead are a piecewise-defined function, piecewise-smooth curve, piecewise linear structure, piecewise algebraic space, etc. They are distinct concepts, not manifestations of some unifying notion. —- Taku (talk) 16:15, 20 July 2024 (UTC)

  In all the case, "piecewise" means that the property that is qualified (way of defining a function, smoothness, being linear, being algebraic, etc.) is true on disjoint open intervals or sets whose union complement has an empty interior. This meaning is sufficiently clear for being understood this way (without further explanation) by experimented mathematicians. So a dab page would be confusing and even disruptive, since a reader that encounters an example that is not among those listed in the dab page should access many articles for trying to find an explantion. In short, a dab page is not convenient for a concept that can be defined in a few lines, and whose all occurences in Wikipedia are examples of instances of this concept. D.Lazard (talk) 18:02, 20 July 2024 (UTC)

  Of course, there is a linguistic meaning of piecewise as piece-by-piece. But it’s not a precise mathematical term, in that you can’t find any textbook defining “piecewise” instead of “piecewise X”. ("continuous” is an adjective too but there is a well-established notion of continuity today if not so in the past.) This seems to be the underlying issue leading to this thread. The purpose of a dab page is that the readers should be pointed to more specific articles, since that’s more helpful than giving an impression that “piecewise “ is a precise mathematical notion. As you said, we may guess the intended meaning of piecewise from the past experiences, but that’s not how we write a Wikipedia article; an article needs to be based on reliable sources, —- Taku (talk) 09:32, 21 July 2024 (UTC)

• Suggestion. A move Piecewise → Piecewiseness would satisfy WP:NOUN. — Vincent Lefèvre (talk) 20:40, 20 July 2024 (UTC)

Piecewiseness instead Vincent Lefèvre. Slightly weak until I make a list of quotes and write a definition for Wiktionary so this becomes a real word (only 168 Google results, but appears on Math.SE). I listed both adjective and adverb examples at wikt:piecewise. With "piecewise-defined function" by Fgnievinski, we need to decide if it has a primary topic of linear piecewise functions or it is a broad-concept article that also covers piecewise polynomial functions. If we collect other things under Related concepts, we could spin off Piecewiseness before, during, after the move, or later.

I don't think "Piecewise property" is a good name. Both set-theoretic properties (associative, commutative, distributive, etc.) and universal properties are external. Piecewiseness is internal. Are there any internal things we call "X property"? The 2nd paragraph of the lede, I think of as "piecewise property property", although sounding unusual. 174.92.25.207 (talk) 00:11, 21 July 2024 (UTC)

• Move to "Piecewise-defined function", as the main concept. Anything else can go under section Piecewise-defined function#Related concepts. fgnievinski (talk) 01:36, 21 July 2024 (UTC)