Make a PHP file with a function

You really shouldn’t be doing this. Functions belong in the functions file. Period. (Or perhaps in a separate functions file which you include in the main one)

However, if you insist, you can drop functions in any template file you want, because PHP doesn’t care. Actually you will need to include it in every template file where you want to call dimox_breadcrumbs(). In which case you would have to be careful not to include it twice in one page. So, only insist if you can handle the hassle.