Proving that a function is not a OWF (One-way-function)
To prove that a function is not a OWF, you can exhibit an inversion algorithm and show that your inversion algorithm succeeds in inverting the OWF with non-negligible probability.  In other words, if we randomly choose $x$, compute $f(x)$, feed $f(x)$ to your inversion algorithm, and let $x'$ denote the output of your algorithm, then you need to show that $\Pr[f(x')=f(x)]$ is non-negligible.  Thus, this is average-case hardness, averaging over all possible inputs to the OWF.