A function $f$ is a worst-case OWF if there is no adversary $\mathcal{A}$ such that $$\forall x,Pr[y=f(x): f(\mathcal{A}(y))=y]=1$$
A weak OWF is a function that the probability of inverting it is bounded by some $1-\frac{1}{q(n)}$ for some polynomial function $q(n)$. The question is suppose $f$ is a weak OWF, can we define a function $g$ that is a worst-case OWF but not a weak OWF?
My idea is to make $g$ easy to invert for (exponentially) most inputs. This make $g$ "weaker than" any weak OWF. For example, consider $$g(x) = \left\{\begin{array}{ll}0 & x < 2^{|x|}-1 \\ f(x) & \text{otherwise} \end{array}\right.$$
Clearly $g$ is weaker than any weak OWF. But how can I show that no adversary can always invert $g$ then?
