The discussion revolves around the concept of continuity of functions on composed intervals, specifically examining whether a function can be continuous on a union of disjoint intervals. The example used is the function f(x) = 1/x and its continuity on the intervals (-∞, 0) and (0, ∞).
Participants present differing views on the interpretation of continuity on composed intervals, with some asserting that continuity can be established over disjoint intervals while others focus on the local nature of continuity. The discussion remains unresolved regarding the implications of continuity in this context.
There are assumptions regarding the definitions of continuity and the nature of the intervals discussed. The implications of continuity in topological spaces and the local versus global nature of continuity are not fully explored.
It's the same. Continuity is a local concept: a function is continuous on some domain D if it is continuous at every point in D. Since your f is continuous at every point in [tex]\mathbb{R}-\{0\}[/tex], it is continuous on every subset [tex]D\subseteq\mathbb{R}-\{0\}[/tex], in particular on [tex]D= (-\infty,0) \cup (0,\infty)[/tex].ronaldor9 said:For example, if [tex]f(x)=\frac{1}{x}[/tex] then on the interval [tex](-\infty,0) \cup (0,\infty), f(x)[/tex] is continous? Or is the function [tex]f(x)[/tex] continuous on [tex](-\infty,0)[/tex] by itself and [tex](0,\infty)[/tex] by itself?