\begin{definition}[Definition:Continuous Real Function/Left-Continuous]
Let $A \subseteq \R$ be an open subset of the real numbers $\R$.
Let $f: A \to \R$ be a real function.
Let $x_0 \in A$. 
Then $f$ is said to be '''left-continuous at $x_0$''' {{iff}} the limit from the left of $\map f x$ as $x \to x_0$ exists and:
:$\ds \lim_{\substack {x \mathop \to x_0^- \\ x_0 \mathop \in A} } \map f x = \map f {x_0}$
where $\ds \lim_{x \mathop \to x_0^-}$ is a limit from the left.
Furthermore, $f$ is said to be '''left-continuous''' {{iff}}:
:$\forall x_0 \in A$, $f$ is '''left-continuous at $x_0$'''
\end{definition}