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Thread: Continuity of functions - Real Analysis

  1. #1
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    Continuity of functions - Real Analysis

    I have a homework assignement due tomorrow morning and I have some idea on how to do this problem but I am not very sure.

    Define functions f and g on [-1,1] by

    $\displaystyle
    f(x) = xcos(1/x), if \quad x \neq 0; $
    $\displaystyle
    f(x)= 0, if \quad x=0;
    $
    $\displaystyle
    g(x) = cos(1/x), if \quad x \neq 0; $
    $\displaystyle
    g(x)= 0, if \quad x=0;
    $

    Prove that f is continous at 0 and that g is not continuous at 0. Explain why these functions are continuous at every other point in [-1,1].

    So I was thinking for f(x) just using the definition of continuity:


    $\displaystyle \left| f(0) - f(x) \right| = \left| 0 - xcos(1/x) \right| = \left| xcos(1/x)\right| < \left| x \right| \left| cos(1/x) \right| < \left| x \right| < \epsilon$
    Choose $\displaystyle \delta = \epsilon $
    $\displaystyle
    \left| x- 0 \right| < \delta
    $
    Ok this proves that f is continuous at 0 right??

    I am not entirely sure how to prove g(x) isn't continuous but also that these functions are continuous at every other points!
    Any suggestion??
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  2. #2
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    Consider the sequence $\displaystyle x_n = \frac{1}{{2\pi n}} \Rightarrow \left( {x_n } \right) \to 0$.
    What is the value of $\displaystyle \left( {\forall n} \right)\left[ {g(x_n ) = ?} \right]$
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  3. #3
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    Ok, so I was able to prove it that g is not continuous at 0 but now I do I prove that f and g are continuous for every x in the interval???
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  4. #4
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    Quote Originally Posted by ynn6871 View Post
    that g is not continuous at 0 but now I do I prove that f and g are continuous for every x in the interval???
    Quote Originally Posted by ynn6871 View Post
    I prove that f and g are continuous for every x in the interval???
    Well you donít do that because it is not true of $\displaystyle g$.
    You proved that $\displaystyle g$ is not continuous at $\displaystyle x=0$ so it cannot be continuous for all $\displaystyle x$.

    You also proved that $\displaystyle f$ is continuous at $\displaystyle x=0$.
    Note that if $\displaystyle x \ne 0$ then both functions $\displaystyle x\,\& \,\cos \left( {1/x} \right)$ are continuous.
    The product of two continuous functions is continuous.
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