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Math Help - continuity question

  1. #1
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    continuity question

    Show that f(x)=x^n where n \in \mathbb{N} is continuous on the interval [a,b].

    using the \delta \ \varepsilon yields:

    \forall \ \delta>0, \ \varepsilon >0 \  \ |x-c|<\delta, \ \mbox{then} \ |x^n-c^n|<\varepsilon for n \in \mathbb{N} at which point I don't know how to continue.

    I figure, I can write f(x)= x\cdot x \cdot x \dotso \cdot x. furthermore I can express this as a number of composite functions where I would have:

    f(x)=x^n=(f \circ f \circ f \circ \dotso \circ f)(x) .

    I know how to show that given two continuous function then their composite is continuous, but I don't know how I would show that n composite functions are continuous, I imagine this would have to be done by induction.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Show that f(x)=x^n where n \in \mathbb{N} is continuous on the interval [a,b].

    using the \delta \ \varepsilon yields:

    \forall \ \delta>0, \ \varepsilon >0 \  \ |x-c|<\delta, \ \mbox{then} \ |x^n-c^n|<\varepsilon for n \in \mathbb{N} at which point I don't know how to continue.

    I figure, I can write f(x)= x\cdot x \cdot x \dotso \cdot x. furthermore I can express this as a number of composite functions where I would have:

    f(x)=x^n=(f \circ f \circ f \circ \dotso \circ f)(x) . This is wrong!

    I know how to show that given two continuous function then their composite is continuous, but I don't know how I would show that n composite functions are continuous, I imagine this would have to be done by induction.
    There's no reason (of possibility) for introducing composite functions here. The function f is the n-th power of a continuous function, and hence is continuous. But this is exactly what you need to prove, so here is a hint for the proof:

    The key inequality is |x^n-c^n|=|x-c||x^{n-1}+x^{n-2}c +\cdots + x c^{n-2} + c^{n-1}|\leq |x-c| n |2c|^{n-1} for any [tex]x[/Math] such that [tex]|x|<2|c|[/Math]. I think you can conclude from there.
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  3. #3
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    The only part I'm a little confused about, is your statement that |x| <2|c|.
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  4. #4
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    Quote Originally Posted by lllll View Post
    The only part I'm a little confused about, is your statement that |x| <2|c|.
    I wrote this for brevity, but in fact, I only need x not to be too large, in order to bound |x|, but not too close to 0 since x is supposed to be about equal to c. For instance, I could have said "Choose any M>|c|. If |x|\leq M, then |x^k c^{n-1-k}|\leq M^k |c|^{n-1-k}\leq A^{n-1} where A=\max(M,|c|,1), so that |x^n-c^n|=|x-c||x^{n-1}+x^{n-2}c +\cdots + x c^{n-2} + c^{n-1}|\leq |x-c|n A^{n-1}. " Then, when you choose \delta, it must be such that \delta n A^{n-1}\leq \varepsilon and \delta\leq M-|c|, so that |x-c|<\delta implies |x|< |c|+\delta\leq M and we can apply the above.
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