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I am thinking of $\lim_{x \to 0} \ {f(x)} = 1$ but I am confused as $x=0$ is not in the domain and also I want to write an $\epsilon$-$\delta$ proof. So any help is much appreciated!

Thanks!

2 Answers 2

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Here is a non $\epsilon-\delta $ proof: Extend $f$ to a function $g:[0,1]\to \mathbb R$ by setting $g(x)=f(x)$ for all $x\in (0,1)$ and $g(0)=1$, $g(1)=\sin (1)$. Now, $g$ is continuous on its domain which is a closed interval and thus is uniformly continuous. It is very easy to show that the restriction of a uniformly continuous function is itself uniformly continuous and thus $f$ is uniformly continuous.

Giving a direct $\epsilon -\delta $ proof will be quite technically messy. Is there any particular reason you wish to obtain such a proof.

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    Because that is how I usually do these proofs and for instance, I don't understand your proof completely because I don't know why the restriction of a uniformly continuous function is uniformly continuous. Also can you explain why g is continuous on the domain?2012-12-16
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    g is continuous at every 02012-12-16
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    you usually do proofs using epsilon delta arguments only in the beginning and for very simple functions. Very soon this becomes too daunting. The whole point of developing a rich theory is that you learn new, more powerful, tools and techniques to proof things. Of course, any proof in analysis can be reduced to a (potentially horribly complicated) argument with epsilon delta. But it helps no-one to see it.2012-12-16
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    i see, yeah i was really struggling to come up with a neat $\ epsilon-\delta$ proof but this makes some sense now. Can I ask you one more thing? Can I conclude f(x)= |x| is uniformly continuous over [-1,1] because its continuous over all R so it must continuous over [-1,1] and since [-1,1] is compact, its uniformly continuous. THanks for the help, i really appreciate it!2012-12-16
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    Yes and you're welcome :)2012-12-16
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    btw, how would you show that $f(x) = sinx/x$ is continuous over (0,1) rigorously? give me an idea and I will try it! THanks!2012-12-16
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    1) the quotient of two continuous functions (when defined) is continuous. 2) sin(x) is continuous everywhere. 3) sinx>0 for all 02012-12-16
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    oh of course and thank you so much!2012-12-16
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    @UH1 how can this proof be extended over the entire $R$ ?2013-10-01
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Ittay Weiss described the right way to solve this problem. If you really want an $\varepsilon$-$\delta$ solution, you can do the following.

Given $\varepsilon > 0$, let $\delta = \varepsilon$. Now for $x,y\in (0,1)$ with $|x-y| \leq \delta$, we have $$|f(x) - f(y)| = |(x-y) f'(\xi)| = |x-y| \cdot |f'(\xi)| \leq \delta |f'(\xi)|,$$ where $\xi \in (0,1)$. Note that $$f'(x) = \frac{x\cos x - \sin x}{x^2}.$$ This function takes negative values on $(0,1)$ since $\tan x > x$ (for $x\in (0,1)$). Also $$\frac{x\cos x - \sin x}{x^2} \geq \frac{x\cos x - x}{x^2} = \frac{\cos x - 1}{x} \geq -1.$$ That is, $f'(\xi) \in [-1,0)$. Therefore, $|f(x) - f(y)| \leq \delta = \varepsilon$.

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    this is nice! appreciate it!2012-12-16
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    Nice solution! Far less messy than I had anticipated.2012-12-16
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    Does this proof hold good if the interval is entire $R$2013-09-19
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    @AmanMittal: Yes,it does. The proof goes through with small modifications. It's no longer true that $f'(\xi) \in (-1,0)$; but it is true that $f'(\xi) \in (-1,1)$.2013-09-19