1, continuous, because lim (x->; 0) {x 2sin 1 ÷ x} = 0 (bounded quantity * infinitesimal = infinitesimal) = f (0)

2. continuous. Because the left limit lim (x->; 0-) {x 2+1} =1= f (0) = right limit lim (x-> 0+) {2^x} = 1

This is the most detailed, because the definition of continuity is:

Left limit = right limit = function value.

This is proved by definition.