I am interested in the statement
Let $A$ be a set and $f_n$ be a sequence of functions defined on $\overline{A}$. Assume that $f_n$ pointwise converges to a function $f$ on $\overline{A}$. If $f_n$ converges uniformly on $A$ to $f$, then $f_n$ converges unifomly on $\overline{A}$.
I initially wanted to use the contraposition of this statement to answer this question. But trying to prove it lead me to think that it may false. But cannot find counter-examples.
My question is then $2$-fold:
- If the statement is false, what is a simple counter-example?
- Under what conditions is the statement known to be true? It is true if $\overline{A}\setminus A$ is finite for example.
I assume no a priori condition on $f_n$ and $f$.
False.
Let $A=\mathbb Q$ and define $f_n(x)=f(x)=0$ for each $x\in\mathbb Q$ and $n\in\mathbb N$. If $x\in\mathbb R\setminus \mathbb Q$, then let $f_n(x)=x/n$ for each $n\in\mathbb N$ and $f(x)=0$. Clearly, $(f_n)_{n\in\mathbb N}$ converges pointwise to $f$ on $\overline A=\mathbb R$. However, convergence is not uniform on $\overline A$, even though it is on $A$.
Incidentally, this counterexample reveals that the additional assumption that the limit function $f$ is continuous (even uniformly continuous) on $\overline A$ is still not strong enough to make the claim true.