Suppose $f(x)=\sin x$ then how can I know in which interval it is uniformly continuous. I can solve easily by prove $|f(x_2)-f(x_1)|<ε$ for $|x_2-x_1|<\delta$ but I can't understand the same function is not uniformly continuous on other interval
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2 Answers
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2 Since $f'$ is a bounded function, $f$ is uniformly continuous on $\mathbb R$ and therefore on every interval of $\mathbb R$.
answered May 18, 2018 at 15:15
José Carlos Santos
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- $\begingroup$ How u know that it is uniformly continuous on R it is also uniformly continuous on [0,∞) $\endgroup$mSourav– mSourav2018-05-18 16:00:43 +00:00Commented May 18, 2018 at 16:00
- 2$\begingroup$ It follows immediately from the definition of uniform continuity that if $A\subset B$ and if $f$ is uniformly continuous on $B$, then it is also uniformly continuous on $A$. $\endgroup$José Carlos Santos– José Carlos Santos2018-05-18 17:17:04 +00:00Commented May 18, 2018 at 17:17
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As $f'(x)= \cos(x)$ is bounded, $f$ is uniformly continuous on the whole of $\mathbb{R}$.
