First, as far as your customer is concerned, it shouldn't be your fault as an individual. Your team screwed up. It's none of their business which individual developer caused the bug.
Second, determine why that bug slipped past you. Didn't you test your code enough? Did you rush this feature because of deadline pressure? Are you working with error-prone code that could be made more robust? Should others have caught the bug in testing?
Everyone writes bugs. That's nothing to feel bad about. The problem is when the bug goes undetected. So who should have detected the bug? Even if you wrote it, it may not necessarily be your fault that it didn't get fixed.
(Which doesn't mean you shouldn't take responsibility, of course. Just that it's not just you, but perhaps the entire team who have to stop and reconsider the process.)
Look ahead. Determine what can be done to avoid it happening again. What should you do while writing code, and what should your company do during testing, or when preparing for a release? What can be done in the code itself to make easier to work with, and harder to screw up?