Let $\sigma$ denote the sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$, let $x,y \in \mathbb{R}$. Given the following two conditions: $|\sigma(-x) - \sigma(y)| < \epsilon$ and $x - y > c > 0,$ where $\epsilon$ can be regarded as a small positive number and $c$ as a large positive number. Can we draw the conclusion that $x > \frac{c}{2} - f(\epsilon), y <-\frac{c}{2} + f(\epsilon)$, where $f(\epsilon)$ is some function of $\epsilon$.