Making git auto-commit

I'd like to use git to record all the changes to a file.

Is there a way I can turn git 'commit' on to automatically happen every time a file is updated - so there is a new commit for every change to a file?

Ideally I'd like my users to not even know that git is running behind the scenes. A user could then potentially "undo" changes to a file - and this could be achieved by pulling a previous version out of git.

On Linux you could use inotifywait to automatically execute a command every time a file's content is changed.

Edit: the following command commits file.txt as soon as it is saved:

inotifywait -q -m -e CLOSE_WRITE --format="git commit -m 'autocommit on change' %w" file.txt | sh

The earlier inotifywait answer is great, but it isn't quite a complete solution. As written, it is a one shot commit for a one time change in a file. It does not work for the common case where editing a file creates a new inode with the original name. inotifywait -m apparently follows files by inode, not by name. Also, after the file has changed, it is not staged for git commit without git add or git commit -a. Making some adjustments, here is what I am using on Debian to track all changes to my calendar file:

/etc/rc.local:

su -c /home/<username>/bin/gitwait -l <username>

/home/<username>/bin/gitwait:

#!/bin/bash
#
# gitwait - watch file and git commit all changes as they happen
#

while true; do

  inotifywait -qq -e CLOSE_WRITE ~/.calendar/calendar

  cd ~/.calendar; git commit -a -m 'autocommit on change'

done

This could be generalized to wait on a list of files and/or directories, and the corresponding inotifywait processes, and restart each inotifywait as a file is changed.