Q:

Pointwise limit of $f_n(0)=0$, $f_n(1/n)=n$, linear in between

What is the pointwise limit of the function $f_n$ which increases linearly on $[0,1/n]$ with
$$
f_n(0)=0,\\
f_n\left(1/n\right)=n,
$$
and is not defined elsewhere, that is, $f_n:[0,1/n] \to \Bbb R$.  
I think the limit is the discontinuous function $f:\{0\} \to \{0,\infty\}$, with
$$
f(0+)=0,\\
f(0-)=\infty,
$$
where
$$
f(0+)=\lim_{x \uparrow 0} f(x). 
$$
But this seems strange. 

A:

Comment transfered to answer by request. 
If you want to talk about pointwise convergence, you have to restrict yourself, to begin with, to the intersection of the domains of your $f_n$'s.  
In this case, this is $\{0\}$.
Now $f_n(0)=0$ for all $n$, so $f_n$ converges pointwise to $0$ on $\{0\}$.

