It is well-known that if f: [a,b]\(\to {\mathbb{R}}\) is Hölder-continuous with exponent \(\alpha\), its graph has Hausdorff dimension at most 2- \(\alpha\) and almost all of its level sets \(f^{-1}\{x\}\) have Hausdorff dimension at most 1-\(\alpha\). The author studies the problem of getting lower bounds using packing measures and dimension instead those of Hausdorff. He shows that if \(\xi\) is a non-decreasing function with \(\xi (0)=0\) and if f satisfies \[ \limsup_{h\to 0}| f(t+h)\quad -\quad f(t)| /\xi (| h|)\quad >\quad 0 \] for almost all t, then the \(\phi\)-packing measure of the graph of f is infinite for any non- decreasing \(\phi\) growing more slowly than \(t^ 2/\xi (t)\). Also if \(\psi\) grows more slowly than t/\(\xi\) (t), the level sets \(f^{-1}\{x\}\) have infinite \(\psi\)-packing measure except for values x in some first category set. Here the first category cannot be replaced by Lebesgue measure zero.