  We investigate the regularity of the positive roots of a non-negative
function of one-variable. A modified H\"older space $\mathcal{F}^\beta$ is
introduced such that if $f\in \mathcal{F}^\beta$ then $f^\alpha \in C^{\alpha
\beta}$. This provides sufficient conditions to overcome the usual limitation
in the square root case ($\alpha = 1/2$) for H\"older functions that $f^{1/2}$
need be no more than $C^1$ in general. We also derive bounds on the wavelet
coefficients of $f^\alpha$, which provide a finer understanding of its local
regularity.
