I cut my hand. How can I be sure that my tendon is not damaged?

See a hand surgeon! A hand surgeon will examine you and perform some tests to make sure that your tendon was not damaged. Sometimes, with partial tendon injuries, you can still bend your fingers. However, if you don't treat it properly, they can tear completely and you will lose finger motion.
See a hand specialis. The best way is to consult a hand surgeon. It is difficult for you to test yourself. In general if you have weakness of the muscles that move your fingers or thumb or you have pain with motion then you may have a tendon injury. If you have any numbness you could have injured a nerve.