We know, that the first period doubling bifurcation oscilates between $\frac{1}{2r}(r+1 \pm \sqrt{(r-3)(r+1)}$. We speak of the logistic map.
We see, that there is a quadratic term. Does that mean, that by changing r with infinite precision, there will be some values that the oscillation will not reach, no matter which r you chose?