sigmoid function

The sigmoid function is a real-valued function of a real variable that takes values strictly between \(0\) and \(1\), approaching each of the two extremes asymptotically. It provides a smooth mapping from the real line to the unit interval and is widely used in analysis and machine learning. Its definition is the following:

\[
\sigma(x) = \frac{1}{1 + e^{-x}}
\]