Consider the function
$$f(x) = \frac{x}{1+|x|}$$
For what we can prove about it's derivative, it exists everywhere, is maximal at $x=0$ ($f'(0)=1$) and we can verify $$\forall x\in \mathbb R : f(x)\in[-1,1]\\\lim_{x\to -\infty} f(x) = -1\\\lim_{x\to \infty} f(x) = 1$$
These are all properties of typical sigmoid functions which are very popular for usage for example in perceptron units used in artificial neural networks, but I have not seen this particular one anywhere in literature, although it is so surprisingly simple.
Is it in use somewhere, or (if not,) is there some particular (mathematical) reason why they are not?