Is there a continuously differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ where
- f(0)=0
- $\lim\limits_{x\rightarrow\infty} f(x) = 1$
- $\lim\limits_{x\rightarrow-\infty} f(x) = 1$
- f(x) increases monotonically when $x\geq 0$
- f(x) decreases monotonically when $x\leq 0$
If possible, I'd be nice to have the rate at which it saturates be tuneable and a double bonus if the function is twice continuously differentiable.
Basically, I'm modeling an efficiency. The process gets more efficient as the magnitude of the input becomes large. As such, I'm looking for a function that can approximate this behavior for a few tests.