An example of uniform convergence on compact sets but not uniform convergence?

Question

As the title suggests, I want to find an example where a sequence of continuous functions $\{f_n\}$ converges uniformly on compact sets to a continuous function $f$, and yet the convergence is not uniform over the whole domain. Thank you in advance!

Answer 1

$x^n$ on $[0,1)$: the compact subsets are contained in $[0,r]$ for some $r<1$.

Other examples are power series with infinite radius of convergence.

Answer 2

Take $f_n:(0,1)\to \mathbb R$, with $f(x)=x^n$.