5 year old fussy eater. At my wits end

[framan]

DD1 is a fussy eater and it is becoming more and more unbearable. She refuses to try new things and when I encourage her to try a taste she adamantly says she hates it. In the end I lose my temper and say that's it there is nothing else to eat. But then I give in and let her eat a banana or some other fruit (which is just about the only healthy thing she will eat. Carrot is her only veg!) And then she will whinge she's hungry and I give in and let her have a piece of bread. I'm just going about it the wrong way I know but just don't know what to do. Her 18 month old sister is on the other hand a fantastic eater and will eat just about anything I give her.
I thought of going to health visitor for advice but my one is not that helpful and a GP might be extreme. What can I do to get her to be more adventurous? Any tips or advice would be more than welcome!
Sorry for the long post but hope you can help.

There is a lot of excellent advice here, thanks. However, I dream of having a child who would even poke at a piece of fruit or vegetable. I don't think I would worry so much about the main meals if I knew that there were healthy snacks I could offer as an alternative. My DS does not eat fruit, vegetables (unless these are in pureed form), cheese, yoghurts...

Mamadothat - don't worry. 2.5 is still too young to reason re: healthy eating, isn't it? We started the candlelit dinners when ds2 was 3 and he could make a promise to try what was on his plate. He's 11 now and will try anything - even tho he doesn't like everything. He will eat any veg, but only really likes apples and bananas - won't eat grapes, strawberries, peaches, citrus etc. Just as long as you are giving pureed fruit & veg your ds will be fine. And don't forget that 'fruit & veg' includes things like raisins & baked beans.