Definition of homogeneous function What's the actual definition of homogeneous function? This link says that the scaling factor should be positive https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/hom/t
Some other resources claim that it can be any real number. Which one is correct and why does it matter?

It depends on the author. I like to call a function $f:V\to W$ between two real vector spaces homogeneous (of degree $1$) if for all $t\in\Bbb{R}$, and all $v\in V$, $f(tv)=tf(v)$.
I would rather reserve the term "positively homogeneous" (of degree $1$) for a function $f:V\to W$ between two real vector spaces such that for all $t>0$ and all $v\in V$, $f(tv)=tf(v)$. But, as I said, it depends on the author whether they want to be picky enough to make the distinction.
The difference between these two concepts can be understood by looking at the following functions $f:\Bbb{R}\to\Bbb{R}$ and $g:\Bbb{R}\to\Bbb{R}$ defined as $f(x)=x$ and $g(x)=|x|$. Then, $f$ is homogeneous of degree $1$, while $g$ is only positively homogeneous (of degree $1$).
