Definition 2

Tag 0BRA, permanent. Cite it as [Bourbaki, Tag 0BRA].

General Topology, chapter IV, § 5. Real-valued functions

Let f be a function defined on a non-empty subset A of $\overline{\mathbf{R}}$ , with values in a topological space X. A limit of f with respect to the filter $\mathfrak{F}$ , if it exists, is called a limit of f on the left at the point a, relative to A, and is denoted by

\lim_{x \to a,\ x < a,\ x \in A} f(x),

or $f(a-)$ , if X is Hausdorff.

Likewise, if $a$ is in the closure of the set $A \cap ]a, + \infty]$ , we define a limit on the right (if it exists) of $f$ at the point $a$ , and denote it by

\lim_{x \to a,\ x > a,\ x \in A} f(x),

or $f(a+)$ , if X is Hausdorff.

The following proposition is an immediate consequence of Theorem 2:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Theorem 2