Clients often want to know if and why an agency, such as ours, charges for the time we take to fix bugs. The misconception that development can happen completely sans-bugs is understandable and common. Ara, our trusty, CTO, just answered this question / concern, with aplomb that we hope was helpful to the client (and possibly to you in your communications on similar fronts):
QUESTION: Are you charging us for bugs that were a result of the development completed by your team?
ANSWER: A good, and common question.
We never charge for negligence, however bugs are another story...
All software contains bugs, all of the time: google chrome, for instance, currently has 59767 open bugs -> https://code.google.com/p/chromium/issues/list.
Web software is especially vulnerable since it targets many platforms: all sites, from Google to Facebook, will also have many bugs all of the time in some browsers. Often the browsers themselves 'fix' these bugs. Sometimes code that worked on Tuesday doesn't on Wednesday due to a version bump - it's a fun little arms race we get to play in the software biz ;-)
So, for sure, bugs are part of software development and no developer can write bug free software all of the time, especially when dependencies such as APIs, ops, variable timelines, and design/ux changes are also factors.
We do try very, very hard to bill clients for functioning software only, and you'll certainly see a few invoice items billed at $0 since we were doing internal meetings/review and/or because they were egregious bugs.
In the end it's a judgement call which I'm happy to discuss in person.