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I Continuity of composition of continuous functions

  1. May 9, 2017 #1
    I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?
     
  2. jcsd
  3. May 9, 2017 #2

    Math_QED

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    Why is ##\log|x|## not continuous?
     
  4. May 9, 2017 #3
    Because of the vertical asymptote at 0?
     
  5. May 9, 2017 #4

    Math_QED

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    But how can you apply the theorem you just stated if ##\log(x)## isn't even defined at ##x = 0##? If you start with false assumptions, surely you will get wrong results.
     
  6. May 10, 2017 #5
    In this theorem you must also trace carefully domains of the functions. The exact formulation is as follows. Let ##X,Y,Z## be topological spaces and ##f:X\to Y,\quad g:Y\to Z## be continuous functions. Then the function ##g\circ f:X\to Z## is a continuous function.
    In your case ##f(x)=|x|,\quad g(y)=\log y,\quad X=\mathbb{R}\backslash\{0\},\quad Y=\{y\in\mathbb{R}\mid y>0\},\quad Z=\mathbb{R}##
     
  7. May 13, 2017 #6

    WWGD

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    But remember that the composition may not even be defined at some points, let alone be continuous. Try specifying the theorem a bit more carefully, as Zwierz , I think, suggested. EDIT: You may want to consider the inverse image of open/closed set is open/closed definition as well.
     
    Last edited: May 13, 2017
  8. May 15, 2017 #7

    mathwonk

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    as suggested above, it is crucial to have a complete and precise statement of a theorem, including hypotheses. in your case the theorem says roughly, if f is continuous at the point p, and if also g is continuous at (and defined on a neighborhood of) the point f(p), then gof is (defined on a neighborhood of and) continuous at the point p. try to see what the points p and f(p) are in your case that give a problem, if any.
     
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