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Is there an $(1)$ infinitely differentiable function that $(2)$ crosses the $x$-axis at and only at every integer where $(3)$ the pattern of the humps' sign is computable but not only by looking at whether every other hump is positive or negative.

For instance, $\sin (\pi x)$ has humps that go ...^v^v^v^v^v... so it doesn't meet $(3)$.

$\sin(\sin(\pi x/2))-\sin(1)\sin(\pi x/2)$ has humps that go ...^^vv^^vv^^vv... so again $(3)$ is the limiting factor.

I was thinking there might be a way to have a function that you can plug in $1$'s and $-1$'s in certain places to make the humps be positive and negative on every interval between integers.

lesath82
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Jacob Claassen
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    What does your third criterion mean? – Kenny Lau May 17 '17 at 03:41
  • the sign of the function is random basically, but due to the nature of the other criteria, it will be humps, like a sine wave – Jacob Claassen May 17 '17 at 03:49
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    The fact that such a function exists should be obvious. Consider two basic buildingblocks and their negatives: high-to-high, high-to-low where these building blocks are some convenient scaling and shifting of a sine function which always touches the $x$-axis on integer values. Paste these together according to some "random"-esque pattern, for example according to whether or not the $n$'th digit of $\pi$ is even or odd (if even, change from low to high or vice versa and if odd stay low-to-low or high-to-high). – JMoravitz May 17 '17 at 04:03
  • your second example touches the $x$ axis at each half-integer, doesn't this disqualify it from consition (2) as well? Just to clarify the rusles, since this would be easily fixed with $x\rightarrow x/2$ – lesath82 May 21 '17 at 22:44
  • Do you mind what happens at negative $x$'s? – lesath82 May 21 '17 at 23:22
  • Negatives matter. And it doesn't really matter what the period is I guess since it can be transformed. – Jacob Claassen May 22 '17 at 00:40

1 Answers1

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Here is your function:

$$ (-1)^{\lfloor \pi \cdot 10^{\lfloor x\rfloor} + \sqrt 2 \cdot 10^{\lfloor -x\rfloor}\rfloor \text{ mod } 10} \cdot e^{-\frac{1}{\sin^2(\pi x)}} $$

where $\pi$ and $\sqrt 2$ in the first exponent have been chosen randomly amongst allegedly normal numbers and can be replaced at will. The first one takes care of the random distribution of humps for positive $x$'s and the other one for negatives.

Obviously the function has to be completed for continuity at each integer $x$.

lesath82
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