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?

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?

• Not to say that this is the way to go, but an alternative is to assume that the limit does NOT exist, and then show that the function cannot be uniformly continuous (i.e., the contrapositive), and it's possible that for this problem, that may be easier. – John Hughes Nov 13 '16 at 12:55