Extending continuity on $\mathbb{R}$ to continuity on $\overline{\mathbb{R}}= \mathbb{R} \cup \{-\infty,+\infty\}$

Question

I am asking myself the following question:

Assume that $(0,\infty)$ and $\mathbb{R}$ are endowed with the euclidean metric repsectively. Let $$f: (0,\infty) \rightarrow \mathbb{R}$$ be a decreasing continuous function. Then the limit $$f(0):= \lim_{x \rightarrow 0, x>0} f(x)$$ exists in $(-\infty,+\infty]$.

Can we then define a topology, or a metric, such that $f$ is continuous as a function from $[0,\infty)$ to $(-\infty,+\infty]$ or to $\overline{\mathbb{R}}=\mathbb{R} \cup \{-\infty,+\infty\}$?

Thanks a lot in advance!