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