\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
deepdream of a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Analysis::Tao::09. Continuous functions on R

Left and right limits

The two separate halves of a complete limit \( \lim_{x \to x_0; x \in X}f(x) \).

Let \( X \) be a subset of \( \mathbb{R} \), \( f: X \to \mathbb{R} \) be a function and \( x_0 \) be a real number.

If \( x_0 \) is an adherent point of [...], then we define the left limit, \( f(x_0-) \) of \( f \) at \( x_0 \) to be:

[...]

If \( x_0 \) is an adherent point of [...], then we define the right limit, \( f(x_0+) \) of \( f \) to be:

[...]

Note: if and only if \( x_0 \) is an adherent point of \( (-\infty, x_0) \) then it is a limit point of \( (-\infty, x_0] \), as adherent to a set minus the element in question is the definition of being a limit point of that set.