TITLE: Bounded function
QUESTION [0 upvotes]: ${f_n}$ is a sequence of continuous functions on $\Bbb R$, and $f_n \rightarrow f$ uniformly on every finite interval $[a,b]$. If each $f_n$ is bounded, is it true that $f$ must be bounded?

REPLY [0 votes]: Answer is NO. For example, Let $f(x)=x$, $f_n(x)=x$ if $-n\leq x\leq n$, $f_n(x)=-n$ if $ x\leq -n$, and $f_n(x)=n$ if $ x\geq n$.
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