# Linearity Of A Function

I understand that the linearity of a function is determined by the degree of the polynomial but I was unsure whether the modulus operator changes this?

Is $f(x) = N \mod x$ a linear function if $N$ and $x$ are integers?

As in:

$f(x) = 17 \mod x$

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That would not be a well defined function. What is $17 \mod 1.23$? – Pedro Tamaroff Apr 14 '12 at 2:07
@PeterT.off Surely OP wants the domain $\Bbb Z$ or $\Bbb N$? – anon Apr 14 '12 at 2:09
@anon I think it is normal to note $x$ a real number. Anyways, it would not be a function like polynomials and linear functions, which the OP mentions, which are usually $\mathbb R \mapsto \mathbb R$. – Pedro Tamaroff Apr 14 '12 at 2:11
Sorry, I didn't think the clarification would change the answer. Both N and x are integers. N is just another variable. It is linear? – Char Apr 14 '12 at 2:15
No. $f(4) = 17 \mod 4 = 1$ but $f(2) + f(2) = 17 \mod 2 + 17 \mod 2 = 2$. – Neal Apr 14 '12 at 2:17

You have to decide what you mean by linear before you can answer this question. The function $f(x)=mx+b$, which you call "definitively linear", satisfies $$f(r-s)-2f(r)+f(r+s)=0$$ for all $r,s$. The function $f(x)=17$ reduced modulo $x$ doesn't: $$f(2)-2f(3)+f(4)=1-4+1=-2\ne0$$ If you want to call it linear, go ahead, but beware that it won't do most of the things that you might expect linear functions to do.
I would say that it is linear over any subset of the domain of the form $(a,a+17)$, where $a$ is where the $17 \mod a = 0$.