IPhone 4 not turning back on????? PLZ HELP or its going into the bin and no more apple

My iphone 4 wont turn on. I used my phone with no prob this morning but then it decided it was turning off and not turning back on again.... I plug it into the wall to charge it tells me to connect to iTunes but as soon as I unplug my phone it turns off. I connect it to computer for iTunes and it restarts after the apple symbol comes up. ???????
I had my phone connected to the computer at work all afternoon and watched it show the apple symbol then switch off and do it again and again. I don't see how it would be the battery as I felt it should have plenty of charge by now plus I have tried all different cables and its still the same.
Please help me. I do like my iPhone but I only seem to have problems.

Those are features of faulty battery. Make Genius reservation and take to Apple Store for evaluation. Or try battery replacement as first step.

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