Does pointwise convergence of a sequence of functions in each closed subinterval of an open interval imply pointwise convergence in the open interval? If a sequence {$f_n$} be uniformly convergent in every closed subinterval $[a+ε,b-δ]⊂(a,b)$, then it isn't necessarily uniformly convergent in $(a,b)$. But what about pointwise convergence? That is, if a sequence {$f_n$} be pointwise convergent in every closed subinterval $[a+ε,b-δ]⊂(a,b)$, is it pointwise convergent in $(a,b)$? {$f_n$} may or may not be uniformly convergent.

Of course it is. Pick a point $x \in (a,b)$. The interval being open gives you an $ \epsilon>0 $ such that $(x-\epsilon, x+\epsilon) \in (a,b)$, and clearly this open interval contains the closed interval $[x-\frac{\epsilon}{2},x+\frac{\epsilon}{2}]$.
