Proving a function is one way hash function
The inverse cannot be calculated, you're right. However, in can be guessed.

In many scenarios, for example cracking hashed password, it's enough for the attacker to know a set of possible inputs which result in certain output (of course if their number is reliable) instead of  one certain value. 

So if the one-to-many function is weak, given the output $y$ the attacker can calculate $x_1,x_2,...,x_n$ such that for each $x_i$ $H(x_i)=y$ and try them one by one.

Concluding - it's not enough to show that it's one-to-many function. You must also show that finding the set mentioned above is computationaly hard for your function.

PS. You might wonder why $n$ can't be so big that trying all possible inputs resulting in a certain output will be impossible. This would cause another weakness - low collision resistance.