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To double jump, jump once, then while in the air, quickly jump again. You will be boosted, and a puff of smoke will spawn underneath you.
To wall jump, hold forward against a wall, while holding jump. You will be propelled off of the wall, and turned 180°. As you are bouncing off a surface, you are able to "double jump" one more time in the air. You are able to do this continuously, double-jumping off of parallel walls, provided they are close enough together.
Showcasing double-jumping and wall-jumping with the Pegasus Boots