Analysis Counterexamples

Question

I am looking for two classes of real sequences of functions to serve as counterexamples:

(1) A sequence of functions $\{f_n(x)\}$ which converge uniformly to some limit function $f(x)$, but each $f_n$ is discontinuous. Possibly some type of step function?

(2) A sequence of functions $\{f_n(x)\}$ which converge pointwise to a limit function $f(x)$ over some totally bounded domain $D$ (i.e. $\forall\, \varepsilon >0$ $\exists \{x_n\}_{n=1}^N$ s.t. $D\subseteq \bigcup_{n=1}^N B(x_n;\varepsilon)$ ), but the convergence $f_n\to f$ is not uniform.

For (2), does

$f_n(x)=n\cdot\chi_{\left(0,\frac{1}{n}\right)}:=\left\{ \begin{array}{lr} n,\quad x\in \left(0,\frac{1}{n}\right)\\ 0,\quad x\in [0,1]\setminus\left(0,\frac{1}{n}\right) \end{array} \right. $

satisfy the requirements, since $\lim_{n\to \infty}f_n(x)=0$ and $[0,1]$ is totally bounded? Either way, do you know of any more straightforward examples?

Thanks in advance!

Answer 1

1. Take $f_n=\frac1n\chi_{\mathbb Q}$.

2. Your example works.

Answer 2

$f_n(x)=x$ if $x\leq 1-\frac{1}{n}$ and $f_n(x)=x+\frac{1}{n}$ if $x>1-\frac{1}{n}$ converging to $f(x)=x$ on the domain $[0,1]$.