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Checking if f(x)=g(x)+h(x) is onto

  1. Aug 9, 2015 #1

    Titan97

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    This is picture taken from my textbook.
    WP_20150809_12_11_28_Pro.jpg
    I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous function may be continuous. (eg: {x}+[x]) So why should h(x) necessarily be a continuous function?
     
  2. jcsd
  3. Aug 9, 2015 #2

    mfb

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    It does not make any statement for discontinuous functions. Sure, there are discontinuous h(x) that would still work, but not in the general case.

    Unrelated:
    The sum of a continuous function and a discontinuous function is discontinuous. Your example sums two discontinuous functions.
     
  4. Aug 9, 2015 #3

    Titan97

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    So h(x) has to be continuous. (I got cinfused while typing). But what makes f(x) onto?
     
  5. Aug 9, 2015 #4

    mfb

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    What do you know if g(x) is a polynomial of odd degree?
    Does the information about continuity and bounds change if you add a bound continuous function?
     
  6. Aug 9, 2015 #5

    Titan97

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    For polynomials of odd degree, the range is (-∞,∞). If h(x) is only defined ∀ x∈[a,b], then f(x)=g(x)+h(x) is only defined ∀ x∈(a,b).
     
  7. Aug 9, 2015 #6

    mfb

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    That's not how "bounded function" is meant here. Its function values are limited, not its domain.
     
  8. Aug 9, 2015 #7

    Titan97

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    The continuity wont change if you add such a function like x+sinx
     
  9. Aug 9, 2015 #8

    micromass

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    Given a polynomial of odd degree ##P(x)##. Think about
    [tex]\lim_{x\rightarrow +\infty} P(x)~\text{and}~\lim_{x\rightarrow -\infty} P(x)[/tex]
    Can those be equal? Can you give an example when those limits will be equal?
     
  10. Aug 9, 2015 #9

    Titan97

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    They wont be equal
     
  11. Aug 9, 2015 #10

    micromass

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    So what will they be concretely? Can you deduce from that that the function ##P(x)## is onto?
     
  12. Aug 9, 2015 #11

    Titan97

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    Yes.
     
  13. Aug 9, 2015 #12

    micromass

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    OK, then the general case shouldn't be too difficult either.
     
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