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Thread: 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 $\displaystyle c \in \mathbb{R}$

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



    Given the following definition for continuity:

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

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


    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, $\displaystyle \infty$)

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

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

    For the ln x, I got down to:

    $\displaystyle |x-a|< \delta$

    $\displaystyle |ln (\frac{x}{a})|$ 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:

    $\displaystyle |x-a|< \delta$

    $\displaystyle |c^a||c^{x-a}-1|<|c^a||c^{\delta}-1|$ 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:

    $\displaystyle |x-a|< \delta$

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

    I thought about using series expansion of sin/cos.
    Last edited by seld; Oct 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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