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Download Gsx Fsdreamteam [2021] Crackl

Download Gsx Fsdreamteam [2021] Crackl




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Show that the function $f(x)=\sin \frac1x$ is neither continuous nor uniformly continuous on $[0,\infty)$


Show that the function $f(x)=\sin \frac1x$ is neither continuous nor uniformly continuous on $[0,\infty)$


Continuity:


Let $f(x)=\sin \frac1x$, we show that $f(x)$ is not continuous on $[0,\infty)$


$f$ is continuous on any interval $(0,r)$ where $0\leq r<\infty$ because $\lim_x\to 0f(x) = 0$ and also as $\lim_x\to 0f(x)= \sin \frac 1 0$ is $0$


$\lim_x\to 0^+f(x) = \sin \frac 1 0$ as $0< \sin \frac 1 0 < 0$


Uniform Continuity:


Let $f(x)=\sin \frac1x$, we show that $f(x)$ is not uniformly continuous on $[0,\infty)$


As $f$ is uniformly continuous on any interval $(a,


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