You need to go into the command prompt, i.e.:
Click start -> Run, type in "cmd" and click 'OK'.
In the black box, type in "ipconfig".
Type in "ping -t (ip address)", with your gateway IP here, as shown below:
You should get a repetitive series of "Reply from ..." lines. Just minimise this black box and use the Internet as you usually would. As soon as you notice the "Work Offline" issue, bring back up this black box. If you see any "Request timed out" messages, then you know the culprit is either with the router or PC's wi-fi.