(1) Let's say we have a series $f = \sum_{n=1}^\infty f_n$ which converges uniformly on every closed interval $[a, b]$ where $a > 0$ is arbitrary and $b$ is fixed. Moreover $f_n$ is continuous for every $n$. Does this imply that $f$ is continuous on $(0, b]$?
I know that uniformly convergent series of continuous functions is continuous. But answers to this questions:
Continuity and differentiability of function series
Continuity of series sumfunction
imply that continuity on $[a, b]$ implies continuity on $(0, b]$ even if series is not uniformly convergent on $(0, b]$. The second question even provided a proof but I am not convinced by it. Because this sequence:
$$f_n(x) = \frac{\sin{nx}}{nx}\cos\left(\frac{x}{n}\right)$$
seems to be uniformly convergent on $[a, \frac{\pi}{2}]$ but not on $(0, \frac{\pi}{2})$. But if I do analogous proof to the one in answer for this question (but for uniform convergence instead of continuity) it shows that this sequence should be uniformly convergent on $(0, \frac{\pi}{2})$.
Could someone explain to me if (1) is true and if that's the case why proof from here can't be extended to uniform convergence, please?