Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

1. Can an entire function be a (pretty) sigmoid?

If so, let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and composition.

2. Does $P$ contain a (pretty) sigmoid?

Clearly, bounded and odd is possible, as in $x \mapsto \sin x$. Bounded and increasing is also possible $x \mapsto e^{-e^{-x}}$.

(Thanks to Kevin Buzzard for noticing that none of the known sigmoid functions are entire functions, my question was originally about $P$ only.)

Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

Let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and product.

If there a function $f$ in $P$ such that $\lim_{x\rightarrow -\infty} f(x) = -\infty$, $\lim_{x\rightarrow\infty} f(x) = 0$ and $e^f - \frac{1}{2}$ is a (pretty) sigmoid.

Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

1. Can an entire function be a (pretty) sigmoid?

If so, let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and composition.

2. Does $P$ contain a (pretty) sigmoid?

Clearly, bounded and odd is possible, as in $x \mapsto \sin x$. Bounded and increasing is also possible $x \mapsto e^{-e^{-x}}$.

Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

1. Can an entire function be a (pretty) sigmoid?

If so, let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and composition.

2. Does $P$ contain a (pretty) sigmoid?

Clearly, bounded and odd is possible, as in $x \mapsto \sin x$. Bounded and increasing is also possible $x \mapsto e^{-e^{-x}}$.

(Thanks to Kevin Buzzard for noticing that none of the known sigmoid functions are entire functions, my question was originally about $P$ only.)

Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

1. Can an entire function be a (pretty) sigmoid?

If so, let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and composition.

2. Does $P$ contain a (pretty) sigmoid?

Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

1. Can an entire function be a (pretty) sigmoid?

If so, let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and composition.

2. Does $P$ contain a (pretty) sigmoid?

Clearly, bounded and odd is possible, as in $x \mapsto \sin x$. Bounded and increasing is also possible $x \mapsto e^{-e^{-x}}$.