Determining whether the sequence ${f_{n}}$ converges uniformly on the set $A$
\begin{equation*} f_{n}(x) = g(nx),\quad g(x)= \begin{cases} x & \text{if }0 \leq x < 1/2,\\ 1-x & \text{if } 1/2 \leq x \leq 1,\\ 0 & \text{if } x > 1. \end{cases} \end{equation*}
I know that the first step is to find the limit of $f_{n}(x),$ which I do not know how, could anyone help me in doing so?
Also the book did not wrote what is the set $A$, can I assume that it is $\mathbb{R}$ and start my solution?
$f_n(\frac 1 {2n})=g(\frac 1 2)=\frac 1 2$ for all $n$. This shows that (while $f_n\to 0$ pointwise) it does not tend to $0$ uniformly. It converges to $0$ uniformly on any set which is contained in $\mathbb R \setminus (0,\epsilon)$ for some $\epsilon >0$. Some details: it is clear that $f_n(x) \to 0$ for each $x$. If $f_n \to 0$ uniformly then, given $\epsilon >0$, there exists an integer $n_0$ such that $|f_n(x)-0| <\epsilon$ for all $x$ for all $n \geq n_0$. The main point here is the same $n_0$ works for all $x$. Even if you make $x$ dependent on $x$ this inequality must hold as long as $n \geq n_0$. To get a contradiction from this inequality you choose appropriate vales of $x$ depending o $n$. This is what I have done. [I get a contradiction when $\epsilon <\frac 1 2$].