Uniform Continuity: How can I show a continuous function $f: ]0,1] \rightarrow R $ is uniformly continuous if $\lim_{x \, \searrow \, 0}f(x) $ exists?

Question

Let $f: ]0,1] \rightarrow R $ be a continuous function. How can I show that $f$ is uniformly continuous exactly then, when $\lim_{x \, \searrow \, 0}f(x) $ exists? I understand this requires a two part answer.
For the first part of the answer, one can show that assuming $\lim_{x \, \searrow \, 0}f(x) $ exists, $f$ can be continued to a continuous function onto the closed interval $[0,1]$. Uniform continuity can then be shown using the theorem that every continuous function $f: [a,b] \rightarrow R $, wherein $[a,b]$ is a compact inteval, is uniformly continuous in that interval.
Thus, for the second part, how can I now show assuming $f: ]0,1] \rightarrow R$ is uniformly continuous, that $\lim_{x \, \searrow \, 0}f(x) $ exists?