Talk:Piecewise

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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)

[edit] 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)

[edit] 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)