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Math Help - continuity of basic functions

  1. #1
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    continuity of basic functions

    I can't figure out how to prove the continuity of the following functions (I've tried searching to see if this question was posted already. I found someone post something about e^x, but no one replied and I also wanted the other functions, so sorry if this is a repeat of a topic already posted):

    given some c \in \mathbb{R}

    \ln x, c^x, \sin x, \text{ and } cos x



    Given the following definition for continuity:

    Let E be a nonempty subset of \mathbb{R} f : E -> \mathbb{R}

    f is continous at a point a \in E iFF given \epsilon > 0 \text{  }  \exists\text{ } \delta > 0 \text{  } s.t. \text{  }  |x-a|< \delta \Rightarrow \text{  } |f(x)-f(a)| < \epsilon


    I have cauchy, sequential characterization of limits, bolzano weirstrauss, squeeze theorem, limit comparison theorems, definition of bounded functions, limit of composite functions [if limit of f exists, and g is continuous, limit (g(f(x)) = g[lim (f(x)) ], extreme value theorem and intermediate value theorem at my disposal.

    My intervals:


    For log, I took my interval as (0, \infty)

    For c^x ( - \infty, \infty)

    For sin/cos ( - \infty, \infty)

    For the ln x, I got down to:

    |x-a|< \delta

    |ln (\frac{x}{a})| but I got stuck there. I thought about raising it to a power of e, but then I'd have to prove that e^x is continuous.


    For the c^x, I got down to:

    |x-a|< \delta

    |c^a||c^{x-a}-1|<|c^a||c^{\delta}-1| but I couldn't find it a way to make it less than epsilon.

    For the sin x and cos x, I got down to:

    |x-a|< \delta

    |sin x - sin a| and cos x, respectively, and I wasn't sure how to even simplify it to take advantage of |x-a|< \delta, much less show that |\sin x - \sin a|< \epsilon

    I thought about using series expansion of sin/cos.
    Last edited by seld; October 23rd 2009 at 09:30 AM.
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  2. #2
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    Were you given this a homework? Normally, these functions are defined in such a way as to make them continuous. What are your definitions of these functions?

    As far as n! is concerned, since n! is only defined for n a non-negative integer, you can't take the limit "as n approaches a" and continuity doesn't really makes sense for such a function.
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  3. #3
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    oh hmm, lol that's true huh. n! goes from N to N.

    Yeah, it's part of an assignment, and I wasn't sure how to go about it. Though the ln x, wasn't I just figured, if I could prove either logarithms were continuous then I could use it to prove that exponential functions are continuous.

    Though that means I could use the series expansions of a^x, and sin/cos as long as I can show that laurent polynomials are continuous.
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  4. #4
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    You should really answer Hall's question: "What are your definitions of these functions?"
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