Infinitely frequently one-sided function based on an assumption on complexity in the mean

We assume the existence of a function $f$ that is computable in polynomial time but cannot be inverted by a randomized average-case polynomial algorithm. The cryptographic setting is, however, different: even for a weak one-way function, a successful adversary should fail on a polynomial fraction of inputs. Nevertheless, we show how to construct an infinitely-often one-way function based on $f$.