A glass broke in my hand and caused a very deep cut. almost can't bend my finger .

Not a DIY job: A cut should be evaluated and repaired while fresh. A "deep" cut is not a Do-It-Yourself job, and one's primary care doctor may not handle "deep" cuts. That means the ER is the best place for care, especially because the ER can call in a hand surgeon or a plastic surgeon if needed to repair blood vessels, muscles, tendons, etc... If unsure, one can call his primary doctor to help decide what to do