The governing equation is $ut=(a(x)ux)x$, $0≤x≤1$, $t>0$, $u(x,0)=0$, $u(0,t)=0$, $a(1)u′(1,t)=f(t)$. The extra data are $u(1,t)=g(t)$. It is assumed that $a(x)$ is a piecewise-constant function and $f≢0$. It is proved that the function $a(x)$ is uniquely defined by the above data. No restrictions on the number of discontinuity points of $a(x)$ and on their locations are made. The number of discontinuity points is finite, but this number can be arbitrarily large. If $a(x)∊C2[0,1]$, then a uniqueness theorem has been established earlier for multidimensional problem, $x∊Rn,n>1$ [see A. G. Ramm, Multidimensional inverse problems and completeness of the products of solutions to PDE, J. Math. Anal. Appl., 134, 211 (1988)] for the stationary problem with infinitely many boundary data. The novel point in this work is the treatment of the discontinuous piecewise-constant function $a(x)$ and the proof of Property C for a pair of the operators ${ℓ1,ℓ2}$, where $ℓj≔−(d2/dx2)+k2qj2(x)$, $j=1,2$, and $qj2(x)>0$ are piecewise-constant functions, and for the pair ${L1,L2}$, where $Lju≔−[aj(x)u′(x)]′+λu$, $j=1,2$, and $aj(x)>0$ are piecewise-constant functions. Property C stands for completeness of the set of products of solutions of homogeneous differential equations [see A. G. Ramm, Inverse Problems (Springer, New York, 2005)].

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