I believe I understand: any algebraic function over rationals can be rewritten as a convergent Puiseux series, and is thus Hölder continuous since the series has only additions, multiplications, and radicals; and polynomially bounded by choosing $n$ large enough.

I don't see how the two claims follow from the convergence of Puiseux series, especially when the polynomial's coefficients must be rational numbers (and not, say, numbers in an algebraically closed field such as the complex numbers). You should edit your answer to give more detail on your proof.

Correction: In the example I gave earlier, the limit is actually 9/7, not 5/4.