I am imagining a family of monotonic functions, which can be represented as a parametric function $f(\cdot; \tau): \mathbb R \to \mathbb R$ parameterized by $\tau \in \mathbb R$ such that (1) when $\tau > 0$, $f$ is increasing, (2) when $\tau < 0$, $f$ is decreasing, and (3) when $\tau = 0$, $f$ is constant.
Furthermore, I want $f$ to satisfy that when $|\tau|$ increases, the strength of monotonicity also increases, i.e., increasing or decreasing faster.
An intuitive example I have in my mind is $f(x; \tau) = \exp(\tau x)$, which is inspired by the softmax function.
Can we have a formal definition for such functions, or are there existing concepts for this?
Any help would be highly appreciated.