Left Function Limit

Last updated Nov 1, 2022

Let $f: \mathbb{R} \to M$ be a Function where $(M, d)$ is a Metric Space and let $x \in \mathbb{R}$. Suppose there exists $y \in M$ s.t. $$\lim\limits_{u \to x} f{\big|}{(-\infty, x]}(u) = y$$ where $f {\big|}{(-\infty, x]}$ is the Function Restriction of $f$ to $(-\infty, x]$. Then we call $y$ the Left Function Limit of $f$ at $x$. We denote it $$\lim_{u \uparrow x} f(u)$$

1. 1. Applying the Heine Criterion to $(-\infty, x]$ gives us the connection between Sequences in $(-\infty, x]$ converging to $x$ and the Left Function Limit at $x$.

• Real Numbers
• Function Limit