As dr jimbob explains, git by default is not quite as secure as you'd probably like for this application. However, there is an option for this sort of situation, which is to sign your commits or tags with GPG.
If you do this, then they have a potentially much stronger guarantee that you have authored this work; you are now relying primarily on GPG and the web of trust to do verification, rather than git itself. The usual GPG things apply, e.g. they need to trust your key, you need to keep your key safe, etc.
Also note that if you sign a particular commit, you are verifying the authenticity of the entire commit history leading up to that one. In your case, this should not be a problem, since you're the only committer to the repository, but it's something to keep in mind while you work.
This may still provide you problems with proving the time. That information is stored in a commit, but you can control that (most obviously through altering your system clock). So what we have at the end of this is a good guarantee that you committed some particular text at what you claim is a certain time. If you want a guarantee that you haven't post-dated data, you will need to involve an external system that has a trusted clock.