Uniform convergence of sequence of functions on every compact subset

Question

I have a sequence $(x_n)$ of continuous functions: $$x_n\colon\mathbb{R}\to\mathbb{R},\quad n\in\mathbb{N}.$$

If it is convergent on $[-a,a]$, $a\in\mathbb{R}_+$, with limit $x$, what do I further need to determine if $x_n\to x$ on $\mathbb{R}$ (According to the sup-norm)? E.g. differentiability?

Is there an applicable statement that is generalizable to Banach-spaces?

Answer 1

Let $I$ denote the identity function, $I(x) = x$. There's no reason to expect that any local condition (such as continuity, differentiability, or real-analyticity) guarantees that if $(f_{n}) \to I$ uniformly on $[-a, a]$ for all $a > 0$, then $(f_{n}) \to I$ uniformly on $\mathbf{R}$.

The sequence $(f_{n})$ defined by $$ f_{n}(x) = x + \tfrac{1}{n} x^{2} $$ confirms the suspicion.

Answer 2

hint

each real $c $ can be put in an interval of the form $[-b,b] $.

thus a pointwise convergence of a sequence of functions $(f_n) $ at $[-b,b] $ implies that $(f_n (c))$ converges.

hence, there will be a pointwise convergence at $\Bbb R $.