Suppose I am given a first order ode of the type
$$\dot x(t) = f(x(t))$$
Then let $t = k$, $k \in \mathbb{N}$
Is it possible to discretize ("Forward Euler") using a time dependent step size $\epsilon(k)$
$$\dot x \cong \dfrac{x(k+1) - x(k)}{\epsilon(k)}$$
$$\implies x(k+1) = x(k) + \epsilon(k)f(x(k))$$
I am asking because I am looking at some discretization of an ODE (pg 5) where $\epsilon(k)$ is used as the discretizing time step. I have only seen the time step taken as a constant i.e. $\epsilon(k) = \epsilon, \forall k$, where $\epsilon$ is small.
Any additional examples will help.