New iPhone 5s urgent please enter and help D

I got a new iPhone 5s last christmas and I started using it the 26th of december, I got a problem that I can't use the 3g function outside which means I can't use internet outsite (I can connect only with Wi-Fi) I tried any solutions suggested from the Apple center's "genius", it wasn't my sim's problem because I tried it on another iphone 5 and it worked perfectly, I did everything possible, now the thing that I can't really stand is arent they supposed to change my phone and give me a new one? Since its a new phone's problem itself? I hate that I payed that much for phone that isn't doing its proper job and all they can do is repair it and not give me a new one?

Added (1). I don't really get your point, your problem might be because you have an old version of phone, I used to have that problem on my iphone 3 and 4 and it's due to the software when the idevices are starting to be slow, people pay a lot to their products it's a choice of mine I don't want people saying then don't buy it, I never encountered this problem before, and its really annoying cause the i've been using the phone in less than 3 months and it was an internal problem and all they can offer is repair?! What the?! I'd get your point if I did broke the phone, like when my boyfriend dropped his iphone 4s inside the pool or if I dropped it on the floor and it was externally broken, there we can talk about your pointless answer.

Added (2). thank you Hugo I did try what you said, but this stupid "genius" sent my phone on repair just yesterday what should I do now? It should be back in 5-10 days can I cancel the repair and complain? I'm really annoyed, I live here in Italy so 4g isn't available here:/

You may have to recover it back to the way you bought it. And begets can't be choosers bro. I mean you don't want them to repair your phone and want a brand new one? You stated how you have a new one. If Apple will repair it, suck it up and take the repair. If that was the case for everyone, then Apple would lose a lot of money. My iphone resets on it's own sometimes and closes apps on it's own. Should I just say I want a brand new one out of the box? Yeah it's a pain but as long as it works and does what it needs to, then you really can't complain.

Why doesn't it work on 4g? Or do you have it set that way? Back up phone on iTunes and hard reset phone back to factory specs. Or I'd try genius bar again. Being only 4 mo's old, they will likely replace at no charge. Had a similar issue few years ago with a 4 and replaced with no hassles.