Is $$\lim_{n \to \infty} x^n $$ a continuous function on [0,1]?
PS: The original question was for $\lim_{n \to \infty} (\sin(x))^n $ but it brought it complications that are not relevant to the main idea.
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2This function does not defiened at $x=(2n+3/2)\pi$ for all $n\in\mathbb{Z}$. – Hanul Jeon Jan 06 '13 at 07:24
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1What is the domain over which you are interested in continuity? Is it the entire $\mathbb{R}$? – Jan 06 '13 at 07:24
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If you add an absolute value sign, the function still remains discontinous – Amr Jan 06 '13 at 07:25
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Arjang: The purpose of this homework is to make you identify the limit pointwise. That is, fix $x$, does the limit exist and what is it? – Did Jan 06 '13 at 07:27
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@did : this is no homework, – jimjim Jan 06 '13 at 07:51
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@tetori : yes, changed the question, thank you. – jimjim Jan 06 '13 at 08:01
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Arjang: The nature of the trouble you had with this question remains mysterious. – Did Jan 06 '13 at 10:54
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@did: Problem was with question: http://math.stackexchange.com/questions/270949/does-bigcap-n-1-infty-frac1n-frac1n-varnothing , I was trying to come up with a question where the statement was true for all n, but in case of limit it broke down. Once I asked a this question : http://math.stackexchange.com/questions/55204/what-are-the-cases-of-not-using-countable-induction . Now what was troubling me with the intersection question was that although 0 is in the intersection for all n, how does one accepts that it is still in the intersection in the limiting case. A simple case for enlightenment. – jimjim Jan 06 '13 at 11:20
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1Thanks for the explanations, unfortunately I fail to see the relevance of this question for the two others, nor the other way round. (If the problem is to find properties which fail to be automatically satisfied when one passes to the limit (whatever that means), try this: every integer $n$ is finite; let $n\to\infty$; then...) Anyway. – Did Jan 06 '13 at 11:33