Sigmoid Function is Strictly Increasing

Theorem

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

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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.

Aiming for a contradiction, suppose, suppose that $S$ is not strictly increasing.

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

Thus:

\(\ds \map S a\)

\(=\)

\(\ds \map S b\)

\(\ds \frac 1 {1 + \map \exp {-a} }\)

\(=\)

\(\ds \frac 1 {1 + \map \exp {-b} }\)

Definition of Sigmoid Function

\(\ds 1 + \map \exp {-a}\)

\(=\)

\(\ds 1 + \map \exp {-b}\)

\(\ds \map \exp {-a}\)

\(=\)

\(\ds \map \exp {-b}\)

\(\ds -a\)

\(=\)

\(\ds -b\)

Exponential is Strictly Increasing and Strictly Monotone Real Function is Bijective

\(\ds a\)

\(=\)

\(\ds b\)

which contradicts $a < b$.

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

$\blacksquare$