Pointwise convergence and continuity problem

Question

I have the following function: $$f_{n}(x)=\frac{n}{1+nx}$$ and want to study its convergence on $[0,1]$.

I let $x$ be fixed on $[0.1]$ and compute the limit of $f_{n}(x)$.

I can rewrite it as $f_{n}(x)=\cfrac{1}{\frac{1}{n}+x}$ which converges to $f(x)=\frac{1}{x}$ as $n\rightarrow+\infty$.

This bothers me however because $f(x)$ is not continuous at $x=0$. (Indeed, if I let $x=0$, then $f_{n}(x)=n$ which converges to $+\infty$ as $n\rightarrow+\infty$).

So do we say that $f_{n}(x)$ does not converge pointwise?

Answer 1

Indeed, since the sequence $\bigl(f_n(0)\bigr)_{n\in\mathbb N}$ does not converge to a real number, the sequence $(f_n)_{n\in\mathbb N}$ is not pointwise convergent on $[0,1]$