Sigmoid Function is Strictly Increasing

Theorem
The real sigmoid function $\map S x$ is strictly increasing.

Proof
By Cumulative Distribution Function of Logistic Distribution, $S$ is the cumulative distribution function of a logistic distribution, with $\mu = 0$ and $s = 1$.

By Cumulative Distribution Function is Increasing, $S$ is an increasing real function.

, suppose that $S$ is not strictly increasing.

Then there are $a < b$ such that $\map S a = \map S b$.

Thus:

which contradicts $a < b$.

Therefore, by Proof by Contradiction, the real sigmoid function is strictly increasing.