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You can test the touchscreen by using diagnostics menu that you'll find under settings > about phone > diagnostics > test the device > touchscreen. If the too of the display doesn't respond I would suggest that you have your phone examined by a service center. Please contact your local Xperia support team for more information:
http://www.sonymobile.com/global-en/support/contact-us/
You could also try to repair the device software usong PC Companion before you turn to the service center:
http://www.sonymobile.com/update
In PC Companion, press start on Support Zone > Start on Update the phone/tablet software > click on the blue highlight (Repair my phone/tablet). Do this without the phone connected. Follow the instructions on screen to perform the repair.
Before repairing the software in your device you may want to backup your information first. Check out this topic for more information on how to:
http://talk.sonymobile.com/t5/FAQ/Backup-amp-Transfer/m-p/407253#U407253
