Let ${f_n}$ be a sequence of bounded continuous functions defined on $\mathbb R$, and suppose that $f_n \rightarrow f$ uniformly on every finite interval $[a,b]$. Is it necessarily true that $f$ is bounded?
For this question, I can easily prove that $f$ is bounded on every finite interval $[a,b]$, but I cannot prove it is bounded on $\mathbb R$. And my professor said that this statement is not true, i.e. $f$ is not necessarily bounded on $\mathbb R$. But I'm having difficulty coming up with a counterexample. Can anyone help?
Best Answer
Consider $f=$ any continuous unbounded function.
Denote by $f_n(x)$ any continuous function such that $f_n(x)=f(x)$ if $x \in [-n,n] $, and $f_n(x)=0$ if $x \in (-\infty,-n-1) \cup (n+1,+\infty)$. It is easy to see that such a function exists, and is bounded (since it is continuous and has a compact support).
Now, using the fact that every interval $[a,b]$ is contained in a interval $[-N,N]$, you can check easily that $(f_n)$ converges to $f$ uniformly on every $[a,b]$ (actually, $f_{n|[a,b]}$ is constant equal to $f_{|[a,b]}$ for $n$ sufficiently large).