Uniform convergence on open interval

Question

Guys can you give me an example of a sequence of functions that is uniformly convergent on R, but not on some open interval (a,b). I am thinking that if a function is it is uniformly convergent on R, and since (a,b) is as a subset of R, then it has to be convergent on (a,b). But I am actually not sure about this. Can you help me? Thank's in advance.

Answer 1

There is no such example. If $(f_n)_{n\in\Bbb N}$ converges uniformly to $f$ on $\Bbb R$, then it also converges uniformly to $f$ on any subset $S$ of $\Bbb R$, whether or not it is an interval. This follows directly from the definition of uniform convergence: If, for any $\varepsilon>0$, there is some $N\in\Bbb N$ such that$$(\forall x\in\Bbb R)(\forall n\in\Bbb N):n\geqslant N\implies|f(x)-f_n(x)|<\varepsilon,$$then, in particular,$$(\forall x\in S)(\forall n\in\Bbb N):n\geqslant N\implies|f(x)-f_n(x)|<\varepsilon.$$