Non-linear function such that $f(ax)=af(x)$ for every $a \in \mathbb R, x \in \mathbb R^n$ > Let $f : \mathbb R^n \to \mathbb R^n$ be a function such that $f(ax)=af(x)$ for every $a \in \mathbb R, x \in \mathbb R^n$. Does it follow that $f$ is linear? I don't think so, if $n>1$. If $n=1$, we have $f(a)=af(1)=\lambda a$ with $\lambda =f(1)$. Otherwise I'm not sure what to do. Thank you!

Wikipedia helps here: For linearity, a function needs to satify two conditions:

* **additivity** (this is missing in your excercise):$$f ( x + y ) = f ( x ) + f ( y )$$
* **homogeneity** of degree 1: $$f ( α x ) = α f ( x )$$



Wikipedia has a nice example of a function with only the second property (The picture on the right side):

$$ f(x,y)=\left\\{\begin{array}{cl} x, & \mbox{if } x \cdot y > 0\\\ 0, & \mbox{else} \end{array}\right. $$

You are right for $n=1$, though. Additivity follows from homogeneity then.