Sequence of functions problem Define
$$
f_n(x)=
\begin{cases}
1-nx, &x\in [0,1/n],\\
0, &x\in [1/n,1]
\end{cases}
$$
Then which of the following is correct:


*

*$\lim_{n\to\infty}f_n(x)$ defines a continuous function on $[0,1]$.  

*$\{f_n\}$ converges uniformly on $[0,1]$.

*$\lim_{n\to\infty}f_n(x)=0$ for all $x\in [0,1]$.

*$\lim_{n\to\infty} f_n(x)$ exists for all $x\in[0,1]$.


I am completely stuck on it. Please help anyone.

I think only 4. is correct. This you can prove by choosing any $x \in [0,1]$. If $x=0$ then $f_n(0)=1$ for all $n$. If $x>0$ then for all $n > \frac{1}{x}$ f_n(x)=0$.
This is one of the standard examples on pointwise but non-uniform convergence, and is probably covered in most analysis courses...
