Hi
Router's IP do not change, there are static.
Computer's IP might change if a regular DHCP is used, and External Internet
IP might be changed periodically by the ISP.
Which specific IP you are looking to monitor?
If you running a Mail server and your ISP is Not blocking the mail server
related ports (many ISP do not allow Mail servers on a none commercial
connection), then you should use Dynamic DNS to maintain a follow up of the
external IP.
Example,
http://www.no-ip.com
Jack (MS, MVP-Networking).