The "Play at home and continue with your portable" idea isn't possible with the GBA due to the differences in technology. The PSP will be much closer to the PS2 and easily allow for this, the problem is you still have to have the game for both formats. Since you can continue between versions both have to play identical. So you'd be paying twice for the same game. I can't see developers (especially bad ones) giving a game two sets of graphics (one for the PS2, one for the PSP), they'll take the short route out and make the home version graphically identical to the PSP one. Since we're likely talking about muliplatform games here, you can expect the PSP to drag down the GC's and XB's version's visuals, too.
No that anybody would buy the PSP, though, its retail price of USD450 is a bit too high.