IPhone 5 won't turn on unless its heated up?

Hi so i've been using my iPhone 5 since it came out and these days it stopped working i have to heat up the back for it to turn on when it get's cold it shuts down i got it to a store they changed the battery and it still has the problem and when it's not hot sometimes the apple logo shows and in about 5 seconds the screen randomizes with collors and fades away so at the store they told me it can't be fixed is it true btw it's just a phone repair store not Apple.

Well if this was my phone, I would have taken it to Apple to get a SECOND (FIRST actually) opinion of the problem.

If Apple tell you it can't be fixed, they will give you options to replace the phone and you will decide if you want to do that through them or some other place selling phones. So check with an Apple store and ask them what your options are being out of warranty as to replacing it.