# Failure functions

## Primary tabs

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\begin{document}
Background:  see  messages.
Abstract  definition:  Let  $\phi(x)$  be  a  function  of
$x$.  Then $x$  =  $\psi(x_0)$  is  a  failure  function  IF  $\phi(psi(x_0))$  is
a   failure  in  accordance  with  our  definition  of  a  failure. Here  $x_0$  is
a  specific  function  of  $x$.   Note  the dual  role  played  by  $x$. It  is  a
variable  in  $\phi(x)$  and  a  function of  a  specific value  of  $x_0$  in $\psi(x_0)$.
\para.
Examples:
\line  1)  Let  our  definition  of  a  failure  be  a  composite  number.Let the
parent  function, $\phi(x)$,    be a  polynomial ring in which  $x$ belongs to
$Z$.  Then  $x = psi(x_0) = x_0 + k*\psi(x_0)$  is  a  failure function.Here
$k$  belongs to  $Z$.

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