He is not lying, he is just talking about something else. When he says 'function' he means a mapping, when he says 'no inverse' he means a mapping that does not map some of the source domain members.
Well, I would have bought that if he had said that his definition of inverse is different from the definition of traditional mathematics.
But he keeps comparing it to traditional mathematics, and in fact, he got his butt handed to him by someone from the Wolfram Alpha team on the Wolfram fora on exactly this quite a few years ago.
Also, if he pretends to know what traditional mathematics is about and where it is wrong, then I can, and will, call him out on that.
If he stops pointing at traditional mathematics and just works on his, what the guy from Wolfram Alpha calls 'hacktower' of logic, work then by all means.
On the talkrational fora, also a few years ago, the fact that he should stop comparing things with traditional mathematics *or* really learn it first was pointed out to him as well.
If you Google for Doron Shadmi then you will find that he has visited a ridiculous amount of math and science fora and on each and everyone this has been said to him.
Also, if you look at the sheer amount of information he *did* look up, then you can not sincerely believe that this has escaped him.
So therefore I can call him a pathological liar, since he knows the difference, he just chooses to act as if we all don't know it and just tries to bluff his way through it.
But you are right in that it is impossible to teach Doron anything, but as I said earlier; Doron is like having a favourite toothache.