Uniform Continuity of sum of a series of functions

Question

Let $f(x)=\sum_{n=1}^{\infty}f_n(x)$, where $$f_n(x)=\begin{cases}n(x-n+\frac1{n}),\ \ x\in[n-\frac1{n},n]\\n(n+\frac1{n}-x),\ \ x\in[n,n+\frac1{n}]\\0,\ \ \text{otherwise}\end{cases}$$

Then, is $f(x)$ uniformly continuous? I think no. But, I am unable to justify through rigour. By looking at the series of functions, I think it is pointwise convergent to zero. But, I dont think it is uniformly convergent. Any hints? Thanks beforehand.

Answer 1

Hints: for your first question, what can you say about $f(n)-f(n+1/n)$? For your second question, what is $f(n+1)-\sum_{1 \leq k \leq n}{f_k(n+1)}$?