# Limit of Uniformly convegent sequence of real valued functions:

I was attempting this qualifying problem. $\{f_n\}_{n=1}^{\infty}$ is a sequence of real valued functions on $\mathbb{R}$. If $f_n$ converges to $f$ uniformly then $f$ must be continuous.

I am little wondering about this problem . I know uniform limit of sequence of continuous functions must be continuous.

If it has to do something with 'real valued functions on $\mathbb{R}$', then I might be missing that trick , otherwise I do not see the reason that the statement is true. But I found this problem as one the past qualifying exam on the website of the university.

Thanks

• Maybe every $f_n$ has to be continuous. Jul 3 '15 at 2:34
• If that was the case , there would not be any problem. I thought the same thing at the beginning. Jul 3 '15 at 2:36
• On the other hand, if it is not the case, there are trivial counter-examples. Jul 3 '15 at 2:37

Take any discontinuous function $f$ and define the constant sequence $f_{n}=f$ for all $n\in\mathbb{N}$. Then $(f_{n})_{n=1}^{\infty}$ converges uniformly to $f$, but $f$ is not continuous. So the claim must be assuming that each $f_{n}$ is continuous.
that's wrong. Take $$f_n(x) = \begin{cases} 1, & x = 0, \\ 0, & x \ne 0. \end{cases}$$ Then, $f_n$ converges uniformly to $f_1$, which is not continuous.
• OP is saying the $f_i$ must be continuous Jul 3 '15 at 2:40