Monico (2024) · Lemma 3.1

# Squashing Function Lab

Explore how a 0-1 squashing function \(\sigma\) separates two distinct points \(x_0\) and \(x_1\).

Lemma 3.1: For any distinct \(x_0, x_1 \in \mathbb{R}\) and any \(\varepsilon > 0\), there exist \(s, t \in \mathbb{R}\) such that \(\sigma(s + t x_0) < \varepsilon\) and \(\sigma(s + t x_1) > 1 - \varepsilon\). The proof solves a 2×2 linear system and uses monotonicity.
\(\sigma(s+tx) < \varepsilon\) \(\sigma(s+tx) > 1-\varepsilon\) \(\sigma(s+tx)\) curve
Squashing function
Parameters
x0 = -1.00
x1 = 2.00
ε = 0.10
Live computation