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Suppose that a function $f(x)$ defined on $[0,1]$ satisfies $f(1/n)\to 0$ as $n\to\infty$. Is it true that $f(x)\to 0$ as $x\to 0^+$? and show that

(a) $f$ is continuous on $[0,1]$ ?

(b) $f$ is differentiable $(0,1)$ ?

I think this will be true case when $f(x)$=$sin(x)$ but I am a bit confused how to prove for any function and how to use $\epsilon$ _ $\delta$ to prove that. Please if any one can help me with this problem.

Thank you.

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For general functions it is false. Example: $f(x)=1$ when $x$ is irrational, $f(x)=x$ when $x$ is rational. – André Nicolas May 2 '13 at 6:08
Please, make a bigger effort to think of a useful title. – Mariano Suárez-Alvarez May 2 '13 at 6:14

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