Regarding Limit/continuity/convergence

Question

let $$f_n(x)=\begin{cases} 1-nx&\text{when }x\in[0,1/n]\\0&\text{when }x\in [1/n,1]\end{cases}$$ Which of the following is correct?

1. $\lim_{ n\to\infty} f_n(x)$ defines a continuous function on $[0,1]$
2. $\{f_n\}$ converges uniformly on $[0,1]$
3. $\lim_{n\to\infty} f_n(x)=0$ for all $x\in [0,1]$
4. $\lim_{n\to\infty} f_n(x)$ exists for all $x\in[0,1]$

Answer 1

Let denote $\displaystyle f=\lim_{n\to\infty} f_n$.

For $x=0$ we have $f_n(0)=1,\quad\forall n>0$ so $f(0)=1$.

For $x>0$ there's $N\in\mathbb N$ such that $\frac{1}{n}\leq x,\quad \forall n\geq N$ so $f_n(x)=0\quad \forall n\geq N$ and then $f(x)=0$ so we conclude: $$f(x)=\left\{\begin{array}\\ 1&\text{if}\ x=0\\ 0&\text{if}\ 0<x\leq1 \end{array}\right.$$

Now can you answer the questions?

It's clear that $f$ isn't continuous at $0$ so options 1. and 2. are false and $f(0)=1$ so also option 3. isn't true and since $f(x)$ exists for all $x\in[0,1]$ so option 4. is true.