Smooth saturation function I need a function similar to 
$$Saturation(x)=min(max(x, -1), 1)$$
except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep the $|Saturation(x)-x|<0.02$ as far as $|x|<0.9$ while $arctan$ gives me a big error. 

The function must be identical around $x=0$ and almost flat after $x=1$.
So, the requirements of the function are:
$$f'(0)=1$$
$$f(0)=0$$
$$f'(1)\thickapprox0$$
$$f(0.90)=0.88$$
$$|f(x)|<1$$
$$|f'(x)| \leqslant 1$$
$$\forall n \in \mathbb{N} , \exists \frac{d^n}{dx} f(x)$$
Can anybody suggest me such a function?

The Logistic Function should do the trick. I set up the version below to capture some of your requirements.
$$\text{Sat}(x)=\frac{2}{1+e^{-2x}}-1$$ 
Note: $\text{Sat}'(0)=1,|\text{Sat}'(x)|\leq 1, |\text{Sat}(x)|\leq 1,\text{Sat}(0.9)\approx 0.72, \text{Sat}(0)=0$ and all derivatives exist.
6 out of 7 isn't so bad...and its not too off at $x=0.9$ (w/in 20%).
Response to OP
If you go the other way around, you would want to find a function:
$$f: f'(0)=1, \lim_{|x|\to \infty} f'(x)=0, \int |f'(x)| dx< \infty, $$
One candidate would be of the form: $f'(x)=e^{-kx^p}, p\in\{2,4,6...\}$ Unfortunately, its integral is not elementary.
