There are some children who produce much more sweat than usual even as infants, which is a condition called hyperhydrosis. Often, they are completely healthy except for this problem.
Sweaty palms and feet is called palmoplantar hyperhidrosis. Excessive sweating is caused by heightened activity in the sympathetic chain which is part of the Autonomic Nervous System. It is located in the chest cavity and has nothing to do with motor or sensory functions. http://www.aafp.org/afp/20040301/1117.pdf
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