Why does a limit and continuity imply uniform continuity?

Question

I have a Satz that I am supposed to prove, though I don't completely understand why the Satz is true.

The Satz states, that if a function $f:[0,1) \rightarrow \mathbb{R}$ is continous and does have a left limit $\lim\limits_{x \rightarrow > 1^{-}}{f(x)}$ it implies, that $f$ is uniformly continous.

It is obvious to me that $\lim\limits_{x \rightarrow 1^{-}}{f(x)} = x_0=\lim\limits_{x \rightarrow 1^{+}}{f(x)}$ means a function is continous at $1$ but why does a limit imply anything about uniform continouity?

I also wonder why my lecturer mentions the left limit so explicitly. Sounds to me like the right limit is completely irrelevant for this statement.

Thanks in advance to the responders :)

Answer 1

Hint: extend $f$ to a continuous function on $[0,1]$.

There is no right limit because the function is not defined for $x > 1$.