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Consider the heat conduction in a nonhomogeneous insulated rod of a unit length, with the ends kept at zero temperature at all times. Our main interest is in the identification and identifiability of the discontinuous conductivity (thermal diffusivity) coefficient a(x), 0 ≤ x ≤ 1. The identification problem consists of finding a conductivity a(x) in an admissible set K for which the temperature u(x, t) fits given observations in a prescribed sense. Under a wide range of conditions one can establish the continuity of the objective function J(a) representing the best fit to the observations. Then the existence of the best fit to data conductivity follows if the admissible set K is compact in the appropriate topology. However, such an approach usually does not guarantee the uniqueness of the found conductivity a(x). Establishing such a uniqueness is referred to as the identifiability problem. For an extensive survey of heat conduction, including inverse heat conduction problems see (Beck et al., 1985; Cannon, 1984; Ramm, 2005) From physical considerations the conductivity coefficients a(x) are assumed to be in

Identifiability of Piecewise Constant Conductivity