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How do we know if a function is infinitely continuous? That mean it is continuous at every point?

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    We prove it. What's the function?2012-05-10
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    A function $f:A\to B$ is said to be continuous on $A$ if for every $\varepsilon>0$, there is a $\delta>0$ such that $|x-a|<\delta$ implies $|f(x)-f(a)|<\varepsilon$ for all $x,a\in A$.2012-05-10
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    If by "infinitely continuous" you are refering to the symbol $\mathcal{C}^{\infty}$, this means that at each point, the function has derivatives of all orders; in particular, it is continuous and differentiable everywhere, the derivative is continuous and differentiable at every point , the second derivative is continuous and differentiable at every point, etc.2012-05-10
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    @QiaochuYuan - Is there any process or algorithm that is for every function on the proof? Or just we have to arrange the usage of theorem or order of theorem for different theorem?2012-05-10
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    @Victor: no. As with many problems, it's possible to encode undecidable problems (e.g. the Halting problem) into the problem of whether some arbitrary function is continuous. In practice these artificial examples don't come up and generally speaking you use simple established facts about continuous functions.2012-05-10

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You shouldn't say "infinitely continuous" for this. A function is continuous if it is continuous at every point of its domain (that is the adopted definition in, say, real analysis).

Going with the definition of continuity of a function $f : D \to \mathbb R$ at a point $x_0$ is a starting point : $$ \forall \varepsilon > 0, \exists \delta > 0 \quad s.t. \forall x \in D, \quad |x-x_0| < \delta \quad \Rightarrow \quad |f(x)-f(x_0)| < \varepsilon $$ but in general one wants to use the fact that most elementary functions are continuous (polynomials, rational polynomials where the denominators don't vanish, fractional exponents, sines, cosines, exponentials, log, and sums/products/compositions of those) because working with the definition all the time will make you crazy. But it's good to be able to work with it though.

Hope that helps,

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    Is it possible that he means absolutely continuous? That is a term used in probability used for distribution functions that have a derivative everywhere and hence a density function exists for it.2012-05-10
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    @Michael Chernick : He said "That means it is continuous at every point." I don't think he means absolutely continuous. Plus, if you check his profile, he's an high school student. Highly unlikely.2012-05-10
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    After I said it I had a feeling that the possiblity was very remote. Is infinitely continuous a real math term for everywhere continuous?2012-05-10
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    @Michael Chernick : Not that I heard of. It sounded like a term one would use when he does not know how to say it and it just gets out the way it feels natural to express it.2012-05-10
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    @Patrick: Since your answer was accepted, you probably did interpret his intent correctly, but he didn't actually say that is was continuous at every point: he made that a question.2012-05-10
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    @Brian : If you looked (or tried to answer) his previous questions, you'll notice he tends to give this kind of ambiguity in his questions ; interpreting in this manner is working quite well for me up to now. =) My answer is not purely mathematical, it is a function of who's asking it. You can't "not" think about this when trying to answer.2012-05-10
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    @Patrick: Of course. That doesn't change the fact that he did not actually make the assertion that you attributed to him. I wasn't objecting to your interpretation; I was merely pointing out that your justification to Michael Chernick is a bit specious.2012-05-10
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    @Brian M.Scott : My comment was precisely telling that if Michael knew how Victor answers (it is not a bad thing not to know, it's just a fact), he probably would've reacted the same as me ; Victor tends to put questions marks when he writes a sentence after a question, whether the next sentence is a question or not. When you know this, you know that the second sentence was an assertion, hence my answer. =)2012-05-10
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    Would maybe the phrase "continuous everywhere" be more apt?2012-07-24