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Definition in my book:

A function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ has bounded support if there exists a closed interval $I$ in $\mathbb{R}^n$ such that $f(x)=0$ if $x \notin I$.

Now I have to show that if $f$ and $g$ have bounded support (in $\mathbb{R}^n$) and $c \in \mathbb{R}$, then $f+g$ and $cf$ have bounded support too.

This is what I did: Since $f$ and $g$ have bounded support, there exist $I_1$ and $I_2$ such that $supp(f)=I_1$ and $supp(g)=I_2$. Then I can state $supp(f+g) \subseteq I_1 \cup I_2$ and $supp(cf) \subseteq I_1$, but that doesn't prove that $f+g$ and $cf$ have bounded support, does it? I believe it only shows that if a bounded support exists, it would a subset of $I_1 \cup I_2$ or more important it would be finite. Any thoughts on the matter? Can the bounded support be an empty set?

Kenneth
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  • Show that the union of two closed intervals is contained in a closed interval (it doesn't have to equal the interval, just be within it). And, yes once you've shown that $supp(cf) \subset I_1$ then you've proved it according to the definition. – Tom Collinge Apr 13 '16 at 13:59
  • $[a, b] \cup [c,d] \subseteq [\min{a,c},\max{b,d}]$ – Paul Sinclair Apr 13 '16 at 14:50

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