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Hi
Dial *#*#7378423#*#* in phone pad > service tests > select touch screen > Use your finger to check every part of the screen like given in the screenshot below.
Alternatively you could download "Xperia Diagnostic" from play store to check the same.
After doing the same, do take a screenshot like above and post it here. If some part of the screen still doesn't respond to your touch then go for a software repair using PC Companion ( steps below) although such problems are always related to hardware issues.
Update / Repair your phone via PC Companion
Download PC Companion and install it. Do not connect your phone until you're asked to do so.
Launch PC Companion > Start Support zone > Start Phone/ Tablet Software update > select Repair my Phone / Tablet > Follow the on-screen instructions it shows.
EDIT: I'm late
, 2 people have already responded while i was typing