worst-case OWF from weak OWF 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 $$g(x) = \left\{\begin{array}{ll}0 &amp; x &lt; 2^{|x|}-1 \\ f(x) &amp; \text{otherwise} \end{array}\right.$$

Clearly $g$ is weaker than any weak OWF. But how can I show that no adversary can always invert $g$ then?
In order to answer this question, you first really need to define what a worst-case OWF is. In particular, is it possible to efficiently sample a hard instance? I assume not, because this would be very hard to formalize and I'm not sure it makes sense. If not, then there is an easy answer:


If there exists a weak OWF then $P \neq NP$
If $P \neq NP$ then let $L\in NP\setminus P$ and let $R_L$ be its associated polynomial-time verifiable relation (i.e., $(x,y)\in R_L$ if and only if $x\in L$).
By the definition of a "worst-case" OWF, take a machine that samples $(x,y)\in R_L$ and outputs $x$. 
The function would be defined to be $f(y)=x$.