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This paper gives a survey of piecewise approximation functions. Aside from the use of dummy variables, little research has employed such functions. Piecewise functions permit the dependent variable of a functional relationship to take on different forms with respect to the independent variable on various subintervals of the domain of the independent variable. Three types of piecewise approximation functions are considered: dummy variables, piecewise linear regressions and cubic splines. The theoretical and empirical properties of such functions are discussed within the context of an illustrative example involving the increase in ridership over time on urban mass transit systems. Dummy variables take account of shifts in the structural relation in a discontinuous fashion. Piecewise regressions and cubic splines allow for structural shifts without discontinuities arising. Tests are performed to determine whether a linear, quadratic, or cubic piecewise continuous function should be employed. In addition a test is employed to see if structural change occurs in the functional relation at the shift points.