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# Pointwise Limits and Uniform Convergence

TOPO-WMFW54

Which of the following sequences of continuous functions $[0,1]\rightarrow[0,1]$ does not have a continuous function $[0,1]\rightarrow[0,1]$ as its pointwise limit?

A

$\{f_n:n\in\mathbb Z_{\ge 0}\}$ where $f_0=\mathrm{id}_{[0,1]}$ and $f_{n+1}:[0,1]\rightarrow[0,1]$ sends:

$$x\mapsto\left\{\begin{matrix}f_n(3x)/2&\text{ if }0\le x \le 1/3 \\\ 1/2 & \text{ if }1/3\le x\le 2/3 \\\ 1/2+f_n(3x-2) &\text{ if }2/3\le x\le 1\end{matrix}\right.$$

B

$\{f_n:n\in\mathbb N\}$ where $f_n:[0,1]\rightarrow[0,1]$ sends $x\mapsto (1-x/n)^n$

C

$\{f_n:n\in\mathbb N\}$ where $f_n:[0,1]\rightarrow[0,1]$ sends $x\mapsto (x/n)^(x/n)$ and, for convenience, $0^0=1$.

D

$\{f_n:n\in\mathbb N\}$ where $f_n:[0,1]\rightarrow[0,1]$ sends $x\mapsto x^n$