Hi to all again, thx for your great comments.
It seems that this game is well known to many readers.

When I formulated my questions I didn't really think about multiple Nash equilibria. I had this one equilibrium in mind many of you pointed out- that x=0 and player two accepts any offer greater or equal to 0.

One of you decribed this nice experiment where he actually played (almost) subgame perfect if the strategy space of player 1 were finite.
But player 1's strategy space in an intervall, hence a continuum.

So the for me striking result is that the above game doesn't have a subgame perfect equilibrium! Why?
(Hint apply Kuhn's lemma and its two critical conditions finiteness and the expected utility hypothesis which can be taken as given.


Solution:
Suppose x=0, accept x >=0 is subgame perfect.
Construct a sequence of player 1's offers which approach 0. What happens to the pay-off function of player 1 in the limit of this sequence? It's discontinuous in the limit! As indifference of player 2 to accept or reject the offer of x=0 can be interpretated as flipping a fair coin.
Weierstrass' Theorem doesn't work, right?

Next suppose player 1 offers x being close to 1 ==> His strategy space isn't compact, hence Weierstrass' Theorem doesn't work again. Player 1 cannot maximize on an open interval.

==> No subgame perfect equilibrium for this game!