Limit of a Real Function at a Point of an Interval

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, 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 $L\in\mathbb{R}$.

1. (The definition.) (i) One says that $f(x)$ tends to $L$ as $x$ tends to $c$ if for every $\varepsilon>0$ there exists $\delta>0$ such that every $x\in I$ satisfying $0<|x-c|<\delta$ satisfies

$$|f(x)-L|<\varepsilon .$$

2. (Notation.) (ii) In that case $L$ is called the limit of $f$ at $c$ and is denoted by

$$\lim_{x\to c}f(x) ,$$

so that one writes $\lim_{x\to c}f(x)=L$.