Talk:Piecewise linear function

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Definition of a linear function

A linear function f(x) is said to be linear if and only if f(αx₁+βx₂) = αf(x₁) + βf(x₂)

According to this definition, the function f(x) = a_Ix + b_I in the article is not linear. Joao 16:51, 1 March 2007 (UTC)[reply]

True, but nonetheless the nomenclature here is standard when used in this context. Michael Hardy 23:04, 1 March 2007 (UTC)[reply]

To be precise about "in this context": It is very common to use "linear" and "affine linear" interchangeably. It is really only in the context of linear algebra (and more generally, module theory) that these terms are strictly distinguished, and in this special case, affine linear is of no interest. In the context where the domain is not an algebraic structure, nominally a group, no one cares. 129.107.225.4 (talk) 05:39, 19 March 2010 (UTC)[reply]

A definition isn't the same as an example. The example is fine for an intro, but there needs to be a section discussing measure 0 sets or simplicial complexes. What exactly is "piecewise?" From Bing's book it requires a triangulation. I.e., locally finite simplices on which the function is affine linear.67.255.14.227 (talk) 04:52, 5 October 2010 (UTC)[reply]

I agree with the OP and I also think it should be stated somewhere that this name "piecewise linear" means that the function is piecewise affine, not linear (even if you call that "piecewise linear" and you define "piecewise linear" to be that). Yes, the current definition implies that it is piecewise affine, and that this is called "piecewise linear", but

it is not said (and should be said) that this means that the function is *piecewise affine*
it is not said (and should be said) that the function is (a priori) *nowhere* a linear function (not even piecewise, i.e., on some subinterval), and none of the affine functions to which it may be equal on some interval, is linear in the sense of the definition [need it to be said? is there another sense?] of what means for a function to be linear. — MFH:Talk 23:38, 1 December 2022 (UTC)[reply]

PS: I have just created the redirect Piecewise affine function. At least that... 15 years later...

Square wave

The square wave is not a function, thus I will be removing it. It fails the vertical-line test, which is when you pass a vertical line through the graph of a function, the line only crosses one point. Still unsure about the sawtooth function. --SaveTheWild (talk) 00:39, 12 November 2019 (UTC)[reply]