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Math Help - Real analysis - continuity of a function

  1. #1
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    Real analysis - continuity of a function

    Hi,

    I do not know how to solve below problem; maybe someone can guide me through it:

    Prove that f(x)=x^n is continuous on R for any positive integer n.

    Thank you.
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  2. #2
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    Re: Real analysis - continuity of a function

    It's easy to show that f(x)=x^n is not only continuous, but uniformly continuous on every closed segment [-C,C]. Indeed, fix a C and assume x_1,x_2\in[-C,C]. We have

    x_1^n-x_2^n =(x_1-x_2) (x_1^{n-1}+ x_1^{n-2}x_2+ \dots+x_1x_2^{n-2}+ x_2^{n-1})

    (this is checked by multiplying the right-hand side). The second factor in the right-hand side has n terms and |x_1^{n-i}x_2^{i-1}|\le C^{n-1}. Therefore, |x_1^n-x_2^n|\le nC^{n-1}|x_1-x_2|. The expression nC^{n-1} is a constant for each fixed n, so a small variation in x produces only a small variation in f(x). It's easy to turn this into an \varepsilon-\delta proof.
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  3. #3
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    Re: Real analysis - continuity of a function

    It also follows from f(x)g(x) is conntinuous if f(x) and g(x) are.
    Thanks from emakarov and Gibo
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  4. #4
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    Re: Real analysis - continuity of a function

    Thank you very much for explaining that. Helped a lot.
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