Let [tex]f[/tex] be a real uniformly continuous function on the bounded(adsbygoogle = window.adsbygoogle || []).push({});

set [tex]A[/tex] in [tex]\mathbb{R}^1[/tex]. Prove that [tex]f[/tex] is bounded

on [tex]A[/tex].

Since f is uniformly continuous, take [tex]\epsilon = m, \exists \delta > 0[/tex]

such that

[tex]|f(x)-f(p)| < \epsilon [/tex]

whenever [tex]|x-p|<\delta[/tex] and [tex]x,p \in A[/tex]

Now we have

[tex]|f(x)| < m + |f(p)| [/tex]

Obviously i should show [tex]b + |f(p)|[/tex] is bounded, but no idea how.

Could someone help me? thx

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# Uniformly continuous and bounded

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