Uniqueness of the Limit of a Real Function at a Point of an Interval

In the setting of The Real Line: Standing Notation and Background for Calculus, let $I\subseteq\mathbb{R}$ be an interval containing at least two points, let $f:I\to\mathbb{R}$, let $c\in I$, and let $L,L'\in\mathbb{R}$.

Then the following hold.

1. (Punctured neighbourhoods are nonempty) ¶ For every $\delta>0$ the set $\{x\in I: 0<|x-c|<\delta\}$ is nonempty.

2. (Uniqueness) ¶ Suppose that $L$ and $L'$ each have the property required of the limit in Limit of a Real Function at a Point of an Interval §limit, that is, for every $\varepsilon>0$ there exists $\delta>0$ such that every $x\in I$ with $0<|x-c|<\delta$ satisfies $|f(x)-L|<\varepsilon$, and likewise for $L'$. Then $L=L'$.

In particular the limit $\lim_{x\to c}f(x)$ of Limit of a Real Function at a Point of an Interval is well defined.