[davidamarquis]

"Define a function h such that h(f(x), f(y)) is true iff x = y"

If an attacker can compute such a function then any cryptosystem would be broken because of the attack you give. The Goldwasser,Micali paper "Probabalistic Encryption" is a very important early result in cryptography about this fact.

The attack implies that no deterministic encryption scheme is secure.

"By our assumptions, the attacker knows f(1)".

There will not be a single value f(1) as secure encryption schemes cannot be deterministic.

There is nothing special about knowing one of the values of f(1) because modern cryptography assumes that the encryption algorithm is public.