Definition:Continuous Real Function/Left-Continuous

Definition
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 $x_0$ the limit from the left of $f \left({x}\right)$ as $x \to x_0$ exists and:


 * $\displaystyle \lim_{\substack{x \mathop \to x_0^- \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$

where $\displaystyle \lim_{x \mathop \to x_0^-}$ is a limit from the left.

Furthermore, $f$ is said to be left-continuous :


 * $\forall x_0 \in A$, $f$ is left-continuous at $x_0$

Also known as
A function which is left-continuous (either at a point or generally) is also seen referred to as continuous from the left.

Also see

 * Definition:Right-Continuous at Point
 * Definition:Continuous Real Function