Does continuously differentiable imply the $\alpha$ order Holder condition for some $\alpha$?

Question

I am reading the book Fourier Analysis written by Stein. The function $f$ satisfies the Holder condition of order $\alpha$ for $\alpha > 1/2$ means that $\sup_{\theta} |f(\theta + t) -f(\theta)| \leq A|t|^{\alpha}$ for all $t$.

Does "continuously differentiable" imply $f$ satisfies Holder condition for some $\alpha > 1/2$?