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.
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 $x$.