(Guidance from :fr:) 
I found a different definition for a linear function in the French Wikipedia. It says that a linear function (fonction linéaire) is a function of the form f(x)=ax (necessarily passes through the zero).
I made further research and I resume here my conclusion on the subject:
A linear function is a function that respects both of the following conditions:
- It must be additive f(X1+X2)=f(X1)+f(X2)
- it must be homogenous f(aX)= a f(x).
the formula you give as a representation of a linear function: f(x)=mx+c is neither additive nor homogenous, hence it is not a linear function although it has a graphical representation of a line.
- Arie Finkelstein
- Hey, what the h*** does homogenous mean? — Preceding unsigned comment added by 126.96.36.199 (talk) 23:55, 10 February 2012 (UTC)
Cleaned up 
So, I stumbled accross this and found it to be rather nasty and hence cleaned it up. The part about definitions being disputed was just plain wrong. The article could use more work, it doesn't really take a look at the mechanics of geometric linear functions very deeply, nd kind of sounds like something from a first year algebra textbook. We should be able to do a lot better than this, we have a lot of great math people on wikipedia. --Matthew 17:23, 14 February 2006 (UTC)
Linear function vs Affine 
Most of the article about linear functions is actually about affine functions. There should be a page on affine functions or first order polynomials and this page should point to those. Somewhere on the page about linear functions it should contain the definition of a linear function: . —The preceding unsigned comment was added by Dsignoff (talk • contribs) 05:25, 10 February 2007 (UTC).
I am finding the same thing. The distinction between linear functions and affine functions is a very important one if you are looking at solving problems with linear algebra.
As an example, if you model a network of electrical resistors with linear functions, you can relatively easily solve for current through them using matrices and linear algebra. If you assume that they are affine functions then the problem becomes much more difficult. Things like lightbulbs and diodes might require affine functions. Marcusyoder (talk) 16:29, 26 August 2012 (UTC)
Relationship between two functions 
Take the following example: Mary has $87 saved, and earns $3.25 per week. Jane has $89 saved, and earns $3 per week. I know there's a relationship between the function 87+3.25x and the function 89+3x, but how do I describe it (besides saying that both functions have a value of 113 when x=8? — Preceding unsigned comment added by 188.8.131.52 (talk) 23:52, 10 February 2012 (UTC)
I suppose the core problem in the accompanying article is simply neglect: too few editors with higher-math knowledge have cleaned up after those who don't realize that their own level of understanding, while it can be part of good and worthwhile articles, is not complete; in some cases, good material may have been discarded as obvious nonsense, by editors who are merely unacquainted with advanced math extensions and generalizations of what they've learned, when we need to cover interrelated topics at both (or more than two) levels.
Compounding that, the structure is fundamentally unsound, and at several levels. Linear function is trying to cover two distinct (tho, yes, related) topics, but says no more about why they appear in the same article than "two different but related concepts". Here's how WP has to deal with such a situation: one article on each of the two (initially, two existing sections become the core of two articles), and a Dab page (or HatNote Dab on one of the articles) to effect navigation to the desired topic/concept's article, since many users don't know how to tell them apart. Once someone can offer the substance behind "related concepts" (instead of just that generic slogan), an article like "Linear-function concepts in mathematics" may be justified; it would discuss the bonds between the two math concepts that are reflected in using (presumably generally in different contexts) the same name for both, and the essential distinctions that render a single concept inadequate. Barring controversy, i will split the accompanying article 4 ways:
- Linear function (disambiguation) (a first line "Linear function may refer to:"; two sub-sentence entries, probably simply 2 links without elaboration,
- Linear function (analytic geometry),
- Linear function (vector spaces), and
- a pathetic stub for Linear-function concepts in mathematics or Linear-function concepts
That much i can do alone, tho collaboration is desirable.
That won't necessarily close the subject: At leastLinearity (which probably needs multiple articles, a Dab, and at least one other umbrella article) must be carefully considered in light of these.
--Jerzy•t 09:56, 23 March 2012 (UTC)
Thanks for your help Jerzy. I took out a lot of the stuff that doesn't actually apply to linear functions. The article is pretty thin right now but at least it isn't wrong anymore. A lot of schools gloss over the actual definition of linearity, but that doesn't mean Wikipedia should too. When I started working on this I read your comment but I didn't realize you had made it so recently. If there is a disambiguation page I think it should note that y = mx + b is not strictly a linear function. The other articles on linear functions also need cleanup. Marcusyoder (talk) 17:02, 26 August 2012 (UTC)