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Let $f$, $g$, $h \colon \mathbf{R} \to \mathbf{R}^\omega$ be defined by

$$\begin{align*} f(t)&:=(t,2t,3t,\ldots),\\\\ g(t)&:=(t,t,t,\ldots),\\\\ h(t)&:=\left(t,\frac{t}{2},\frac{t}{3},\ldots\right) \end{align*}$$

for all $t \in \mathbf{R}$. How to determine whether these functions are continuous relative to the uniform topology on $\mathbf{R}^\omega$? We of course assume that $\mathbf{R}$ is given the usual topology.

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What have you tried so far? And is it a homework? –  Thomas E. Feb 7 '13 at 21:47

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  1. Let $U=\{x\in\Bbb R^\omega:\|x\|<1\}$, where $\|x\|=\sup_{n\in\omega}|x_n|$. $U$ is open in $\Bbb R^\omega$. (Why?) What is $f^{-1}[U]$? Is it open in $\Bbb R$?

  2. Show that for any $s,t\in\Bbb R$, $\|g(s)-g(t)\|=|s-t|$; use this to show that $g$ is continuous. In fact you can use it to show that $g$ is even a homeomorphism.

  3. Show that for any $s,t\in\Bbb R$, $\|h(s)-h(t)\|=|s-t|$.

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