# Can a function be continuous on a composed interval?

1. Jan 27, 2010

### 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)?

2. Jan 27, 2010

### mathman

Re: Continuity

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.

3. Jan 27, 2010

### Landau

Re: Continuity

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 $$D= (-\infty,0) \cup (0,\infty)$$.

4. Jan 28, 2010

### Norman.Galois

Re: Continuity

Great question with great answers!