  The governing equation is $u_t = (a(x)u_x)_x$, $0\le x\le 1$, $t>0$,
$u(x,0)=0$, $u(0,t)=0$, $a(1)u'(1,t)=f(t)$. The extra data are $u(1,t)=g(t)$.
It is assumed that $a(x)$ is a piecewise-constant function, and $f\not\equiv
0$. 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 $a(x)\in C^2[0,1]$, then a uniqueness theorem has been established earlier
for multidimensional problem, $x\in \mathbb{R}^n, n>1$ (see MR1211417
(94e:35004)) 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 $\{\ell_1, \ell_2 \}$, where $\ell_j:= -\frac{d^2}{dx^2} + k^2
q_j^2(x)$, $j=1,2$, and $q_j^2(x)>0$ are piecewise-constant functions, and for
the pair $\{L_1, L_2 \}$, where $L_ju:=-[a_j(x)u'(x)]'+\lambda u$, $j=1,2$, and
$a_j(x)>0$ are piecewise-constant functions. Property C stands for completeness
of the set of products of solutions of homogeneous differential equations (see
MR1759536 (2001f:34048))
