Definition:Continuous Real Function/Left-Continuous/Point

From ProofWiki
Jump to navigation Jump to search

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 $x_0$ if and only if 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 if and only if:

$\forall x_0 \in A$, $f$ 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


Sources