Yes, its possible, but be aware that not all ISP let you do this. For example, i know comcast does not allow web servers in you broadband connection. I personally have used AT&T Yahoo DSL with Static Ip for $64 dollars a moth, its been working fine, i've never had any issue, its pretty reliable.
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if you have dynamic (meaning your ip changes every so often) you can still do it, but i just requires more maintenance from you because you have to change your ip everytime you ISP DHCP server changes your ip.
When i first began to mess around this idea of running some of my websites from home, i connected a LINUX Apache server to my Broadband Cable Connection it worked fine. I switched to SBC YAHOO DSL, but now its called AT&T YAHOO DSL, i've had it for more than a year and, during which i had to move from my apartment and the transition was ok. I didn't have any major problems, i was able to keep the same IPs. so it worked out.
These are some of the question i often get asked:
How do I host my own web site at home?
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I have a DSL connection running into my home, is it possible for me to become a web hosting provider?
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How do I set up a DMZ for safer home web hosting?
Should I host my own web server?
Now Run Your Own Web Server From DSL With A Domain Name
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