Assuming continuity of $f$, how does one prove uniform continuity of $f$, assuming the limit to $+\infty$ and $-\infty$ is $0$?

Question

Assuming continuity of $f$, how does one go about proving uniform continuity of $f$, assuming the limit to $+\infty$ and $-\infty$ is $0$?

Note: $f\colon\mathbb R\to\mathbb R$.

Looking for hints, not solutions. Thanks. I'm having trouble getting from a closed interval being bounded, to the entire function being bounded.

Answer 1

Hint: Use the fact that continuity implies uniform continuity on any compact set, find a suitable set $[-N,N]$ and a patching overlap set $[N-1,\infty),(-\infty,-N+1)$ to get overlapping uniform continuity