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?
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