Explanation of $\frac{sgn(x) + 1}{2} \approx sigmoid(x)$

Question

I saw an approximation in my textbook which states

$$ \frac{sgn(x) + 1}{2} \approx sigmoid(x)$$

Why is this true? Is there any famous reference to where this approximation came from or even some kind of derivation/proof? As I cannot find a relationship, one is continuous and one is not.

Here is a plot of sgn(x)

And here is a plot of sigmoid(x). I can see they are similar at $y=0,1$ but not in between, as there is a jump.

Answer 1

For positive $x\gg1$, $e^{-x}\ll1$ so $\sigma(x)\approx1-e^{-x}\approx1$. For positive $-x\gg1$, use $\sigma(-x)=1-\sigma(x).$ Small $x$ satisfy $\sigma(x)\approx\sigma(0)=\tfrac12$, in line with the Gibbs phenomenon for a Fourier treatment of step functions.