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Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$.

Assume that $\epsilon\in (0,1)$; does $\{f_n\}$ converges uniformly on $[1-\varepsilon,1]$? How about on $[0,1-\varepsilon]$?

My idea is checking whether $f_n$ is continuous on the interval above, but it seems difficult. Can someone give me any idea?

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