Let $f: [0, \infty) \to \mathbb R$ be a continuous function.

We say that $g: [0, 1] \to [-\infty, \infty]$ is an **infinity limit** of $f$ if there exists a sequence of positive reals $r_n$ monotonically increasing to $\infty$ such that $f_n: [0, 1] \to [-\infty, \infty]$ defined by $f_n (x) := f(r_n x)$ converges pointwise to $g$, and the convergence is uniform on $[\epsilon, 1]$ for every $\epsilon > 0$.

For example, a function that approaches a limit $L$ at infinity has the constant function $L$ as it’s only infinity limit.

For a continuous function $f$, is the set of infinity limits of $f$ at most countably infinite?