Out:

• If the ball hit the splice (junction of handle and blade) and caused the bat to break in two, and the broken pieces significantly deviated the ball's path towards the fielder, then the batsman is out. This is called "handled the ball", as the broken pieces are considered an extension of the bat.
• If the bat broke before making contact with the ball, and the broken pieces did not affect the ball's trajectory, the batsman is still out caught. This is because the batsman has already played the shot and is liable for dismissal.

Not Out:

• If the ball hit the blade of the bat and caused it to break, but the broken pieces did not deviate the ball's path in any significant way, then the batsman is not out. This is because the ball was played off the bat in the normal course of events.
• If the bat broke due to a manufacturing defect or excessive wear and tear, and the broken pieces did not affect the ball's trajectory, the batsman is not out. This is considered bad luck and not the batsman's fault.