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Math Help - 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

     <br />
f(x) = xcos(1/x), if \quad x \neq 0;
    <br />
f(x)= 0, if \quad x=0;<br />
     <br />
g(x) = cos(1/x), if \quad x \neq 0;
    <br />
g(x)= 0, if \quad x=0;<br />

    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:


    \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  \delta = \epsilon
     <br />
\left| x- 0 \right| < \delta<br />
    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 x_n  = \frac{1}{{2\pi n}} \Rightarrow \left( {x_n } \right) \to 0.
    What is the value of \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 g.
    You proved that g is not continuous at x=0 so it cannot be continuous for all x.

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