Definition of a non-linear function

Question

I would like to use in a proof that $f(x)$ is a non-linear function of $x \in \mathbb{R}$, without assuming that $f$ be differentiable/analytic. I can also use that $f(0)=0$.

Q: What is a convenient definition for this?

If $f$ were differentiable, I could use something along:

$f(x)$ is non-linear in $x$ if $\exists$ $x_1$, $x_2$ such that $f'(x_1) \neq f'(x_2)$.

The best I can come up with for a non-differentiable $f$ is:

$f(x)$ is non-linear in $x$ if $\exists$ $x_1,x_2,x_3 \neq 0$ such that $\frac{f(x_2)}{f(x_1)} \neq \frac{f(x_3)}{f(x_1)}$. (There must be at least one $x_1 \neq 0$ since otherwise $f(x)=0$ and thus not non-linear.)

Is there any standard definition out there that I am missing? Any pitfalls I am overlooking?

Edit: By linear function I mean that e.g. $g(x)$ is linear if $\forall x \in \mathbb{R} \quad g(x)=\kappa x$ where $\kappa$ is a constant.

Answer 1

Your definition:

Let $f: \mathbb{R} \to \mathbb{R}$ be a function.

$f$ is called linear, iff there exists $k \in \mathbb{R}$ such that for all $x \in \mathbb{R}: f(x) = kx$

or written in quantifier notation:

$f$ is called linear $\iff \exists k \in \mathbb{R}: \forall x \in \mathbb{R}: f(x) = kx$

By taking the negation of this definition, we obtain:

$f$ is not linear $\iff \not\exists k \in \mathbb{R}: \forall x \in \mathbb{R}: f(x) = kx \iff\forall k \in \mathbb{R}: \exists x \in \mathbb{R}: f(x) \neq kx$

Answer 2

You could use:

$$\frac{f(x_2)-f(x_0)}{x_2-x_0} \neq \frac{f(x_1)-f(x_0)}{x_1-x_0}$$

for some $x_2, x_1, x_0$

This should be valid also for general linear function: $g(x)=ax+b$

Answer 3

Since $f$ being linear in the provided sense means that $f(x)=kx$ then you know that $f(a)/f(b)=a/b$ for all non-zero $b$.

You can for instance say that a function $g$ is non-linear if $g(a)/g(b)\not = a/b$ for some pair $a,b$.

(This is precisely the $x_0=0$ case of the other answer, and essentially the same as MathQED's comment.)