Uniform Convergence of $\sum f_n$ if $f(\sum f_n \to f)$ is uniformly continous

Question

Consider a series of functions $\sum f_n$ on an interval $(a,b)$ , such that :

1.$\sum f_n$ converges to $f$ pointwise

2.Both $\sum f_n$ and $f$ are continuous on $(a,b)$

Then if $f$ is uniformly continuous on $(a,b)$ then can we say that $\sum f_n$ is uniformly convergent on $(a,b)$

This is not a question in any textbook , it is just a thought I got when reading about uniform convergence of series of functions.

Is this statement true , or are there any similar results regarding uniform convergence of series of functions ??

Answer 1

No.

Take $f_n=x^n$ on $(0,1)$

Then $f_n \to 0$ pointwise and not uniformly.

You can see this sequence as a series: $$f_n=f_1 +\sum_{k=1}(f_{n+1}-f_n)$$

Answer 2

Take any sequence of continuous functions $(g_n)$ converging point-wise to $0$ on $[a,b]$but not uniformly. Take $f_1=g_1$ and $f_n=g_n-g_{n-1}$ for $n \geq 2$. This gives a you a counter-example.