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Definition: Let f and g be two paths with a common domain [a, b] and D be a subset of C containing the graphs of f and g. If, for each t in [a, b], the line segment joining f(t) and g(t) lies in D, then f(t) and g(t) are homotopic in D because the function h(s,t) = sg(t) + (1-s)f(t) serves as a homotopy. In this case we say that f and g are linearly homotopic in D.
Source: Mathematical Analysis, second edition by Tom M. Apostol