Can a function be continuous on a composed interval?

[ronaldor9]

Can a function be continuous on a composed interval? For example, if f(x)=\frac{1}{x} then on the interval (-\infty,0) \cup (0,\infty), f(x) is continous? Or is the function f(x) continuous on (-\infty,0) by itself and (0,\infty) by itself (If you don't get what I'm trying to say reply back)?

 

[mathman]

If you study the general definition of continuity, using topological spaces, there is no requirement that the domain be a connected set. So in general, the answer to your question is yes.

 

[Landau]

For example, if f(x)=\frac{1}{x} then on the interval (-\infty,0) \cup (0,\infty), f(x) is continous? Or is the function f(x) continuous on (-\infty,0) by itself and (0,\infty) by itself?

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 \mathbb{R}-\{0\}, it is continuous on every subset D\subseteq\mathbb{R}-\{0\}, in particular on D= (-\infty,0) \cup (0,\infty).

 

[Norman.Galois]

Great question with great answers!