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Definition: Two paths f and g. mapping the interval I=[0, 1] into X, are said to be path homotopic if they have the same initial point x_0 and the same final point x_1, and if there is a continuous map F: IxI -> X such that F(s, 0)=f(s) and F(s, 1)=g(s), F(0, t)=x_0 and F(1, t)=x_1, for each s in I and each t in I. We call F a path homotopy between f and g.
Source: Topology (second edition) by James R. Munkres