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Can I express the continuity on multiple intervals using union? For example, I want to discuss the continuity of $f(x)=\frac{1}{x}$. Can I say that it's continuous on $(-\infty,0)\cup(0,+\infty)$ rather than on $(-\infty,0)$ and $(0,+\infty)$? In my opinion, the function does have a continuity at every point of the interval union so I can say it's continuous.

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  • $\begingroup$ Yes. $f$ is continuous at every real number $x$, except $x=0$. Another way of saying this is $f$ is continuous on the union you have written, which by the way is not an interval, rather a union of intervals. $\endgroup$ – Matematleta Jun 16 '15 at 11:48
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Yes, this is possible, although $(-\infty,0)\cup (0,\infty)$ is not an intervall. One can show that a rational function $f:D\rightarrow\mathbb R$ is continuous at every point of its domain $D\subset\mathbb R$ und thus one can say that it is continuous on $D$. Here one has $$f:\mathbb R^*\rightarrow\mathbb R,x\mapsto \frac 1x$$ with $\mathbb R^*=\mathbb R\setminus\{0\}=(-\infty,0)\cup (0,\infty)$.

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