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Definition: If f is a path in X from x_0 to x_1, and if g is a path in X from x_1 to x_2, we define the product f*g of f and g to be the path h given by the equations h(s)=f(2s) if s is in [0, 1/2] or g(2s-1) if s is in [1/2, 1]. The function h is well-defined and continuous, by the pasting lemma, it is a path in X from x_0 to x_2. We think of h as the path whose first is the path f and whose second half is the path g.
Source: Topology (second edition) by James R. Munkres