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?
 A: Normalizing $x$ by the $L^p$-norm of $(1, x)$ would work.
$$L^p(\vec{x}) = \sqrt[\Large p]{\sum_i|x_i|^p}$$
Your smooth saturate would then be this:
$$Sat(x) = \frac{x}{\sqrt[\Large p]{1+|x|^p}}$$
As $p$ approaches infinity, you'll more closely approximate the original $Saturation$ function because $L^\infty$ is equivalent to max. Furthermore, as $p$ approaches $0$, $Sat(x)$'s second-order derivative is larger around $0$, so $Sat(x)$ will more quickly not approximate $y=x$.
Since $p$ and $x$ are real and p is positive, $Sat^\prime(x)$ is well-defined as (from WolframAlpha):
$$Sat^\prime(x) = \frac{1}{\left(1 + |x|^p\right)^{\large \frac{p}{p+1}}}$$
To meet your exact conditions, if you use $p\approx11.56$, then $Sat^\prime(1)\approx0.4709$ (about as good as the tanh suggested by user76844) and $Sat(0.9)\approx0.8800$. All higher-order derivatives should exist, but it would be generally easier to calculate them numerically.
A: 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.
A: My favorite saturating function is tanh(x). Limits nicely at plus/minus 1. If you need more gain in the linear region, modify it as in tanh(gain*x).
