  Singularities of the Radon transform of a piecewise smooth function $f(x)$,
$x\in R^n$, $n\geq 2$, are calculated. If the singularities of the Radon
transform are known, then the equations of the surfaces of discontinuity of
$f(x)$ are calculated by applying the Legendre transform to the functions,
which appear in the equations of the discontinuity surfaces of the Radon
transform of $f(x)$; examples are given. Numerical aspects of the problem of
finding discontinuities of $f(x)$, given the discontinuities of its Radon
transform, are discussed.
