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Is it true that any nonzero function $f: \mathbb R \to \mathbb R$ which is either:

1) constant, or
2) a polynomial, or
3) $\exp$, or
4) $\log$, or
5) any finite combination of the above using addition, subtraction, multiplication, division and composition, (and individually considered as functions from $\mathbb R$ to $\mathbb R$),

Has a finite number of zeros?


Another attempt. 

I'm actually interested in asymptotic behavior of functions, so a function f(x), as far as I'm concerned, is any expression constructed using the syntax below, such that starting from some point c>0, f(x) is defined for all x>c and takes real values everywhere in this range. A function has the following syntax:

 F --> real number F --> exp F --> ln F --> -F F --> F + F F --> F(F) 

Hope this specifies exactly what I mean and excludes everything I DON'T. Any help is appreciated.

  • 2
    The constant function $0$ has infinitely many zeroes...2011-10-21
  • 2
    The constant zero function has an infinite number of zeros.2011-10-21
  • 0
    $1-(x^2/2)+(x^4/24)-(x^6/720)+\dots$ has infinitely many zeros.2011-10-21
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    Let me clarify. The function is not allowed to be identically zero, and in (5) it must be a finite expression.2011-10-21
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    Ok, how about $f(x) = \exp(ix)+\exp(-ix)$?2011-10-21
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    Only real numbers are allowed2011-10-21
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    You said a "real(-valued) function"...maybe you should edit your question to make explicit the conditions you wish to impose.2011-10-21
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    This seems like it's true (when restricted to the *reals*). exp and ln are strictly monotonic. Any strictly monotonic function will have at most 1 zero. Also, non-constant polynomials are piecewise monotonic. So this seems plausible. I'm not sure how I'd go about proving it though.2011-10-21
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    $e^{e^{\frac{\log (-x)}{2}}}+e^{-e^{\frac{\log (-x)}{2}}}$ has infinitely many zeroes.2011-10-21
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    Let's not be deliberately obtuse here. It's not hard to figure out that the OP asks us to consider the smallest set of partial functions $\mathbb R\to\mathbb R$ that contains all constant real functions, the exponential function, and the logarithm, _and_ is closed under composition, addition, negation, multiplication, and reciprocals. (That means no limits and no complex intermediate values. We get polynomials for free since $\log\circ\exp$ is the identity). He then asks whether this set contains a nonzero function with infinitely many zeros.2011-10-21
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    @J.M.: How so? The two exponentials will be positive where defined, so how can their sum be zero?2011-10-21
  • 0
    It can by using negation. Ignore.2011-10-21
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    This question must be rephrased. The logarithm is not a function from $\mathbb{R}$ to $\mathbb{R}$.2011-10-21
  • 0
    I rephrased it. Please take another look.2011-10-21
  • 0
    @Arturo: That was more or less a rewrite of cardinal's comment.2011-10-21
  • 0
    We must also avoid things like $x+(-x)$.2011-10-21

2 Answers 2