The error message:

Fatal error: Call to undefined function update_core()

The Cause:

You don’t have enough disk space on your hosting account anymore.

The Fix:

Pick one:

• Ask your web host to bump up your allowed disk space
• Delete unneeded files on your account
• Move to a new web host

Conclusion

I never ran into this problem before, but during an update to WordPress 3.4, the error kept appearing every time the upgrade script tried to unpack the downloaded zip file from wordpress.org.  It would be helpful if they included a message about it to let you know you may be out of space, but at least you can Google the error and find this page now 🙂