Is my child theme working properly

There’s a simple way to test:

• add wp_die('it works'); to functions.php, if you see it works on every page, then it’s worked
• add a silly CSS rule to style.css, e.g. body { border: 100px solid pink; }, if your pages now have a 100 pixel pink border then your stylesheet is working

If it works, then you did it correctly, if you didn’t do it the correct way, it wouldn’t work.

Otherwise, the only thing’s I’d note are that:

• it’s functions.php not function.php
• avoid closing PHP tags in functions.php, PHP will auto-add them if needed, and it prevents a common mistake with whitespace after the tag
• my_theme is a very generic prefix, you should pick something more unique