How can we prove that the sigmoid function is not homogeneous?
For the sigmoid function $$S(x) = \frac{1}{1+e^{-x}} $$ I want to show that it does not satisfy $$ \forall (\alpha, x) \in \mathbb{R}^+ \times \mathbb{R}, S(\alpha x) = \alpha S(x) $$
I can get to $$ \frac{\alpha}{1+e^{-x}} = \frac{1}{1+e^{-\alpha x}} $$ which obviously is true for $\alpha = 1$ but it is trivial to find a counterexample e.g. $\alpha = 2, x = 1$ ... but is there a nicer way to show that it doesn't hold?
Another easy choice of parameter is to let $x=0$, and you can let $\alpha$ be any positive number not equal to $1$.
Alternatively, you can rearrange them such that $$\alpha\left( 1 + e^{-\alpha x}\right)=1+e^{-x}$$
and we can see that the LHS is dependent on $\alpha$ while the right hand side is not. If you let $\alpha \to 0 $, the LHS goes to $0$, but the right hand side is at least $1$. We can also let $\alpha \to \infty$ to get a contradiction as well.