Let X be a random variable with Cauchy distribution. The author describes a class of functions f(x), \(x\in R\), possessing the following property: The random variables X and f(X) are of the same type, i.e. \(P(f(X)0.\) A generalization of a result of Knight is obtained: If the random variables \(\xi\) and \[ k\xi +\alpha -\sum^{m}_{j=0}p_ j/(\xi - \gamma_ j), \] where \(k\geq 0\), \(\alpha\in R\), \(P_ j>0\), \(\gamma_ 00\) the distribution of \(a+pX\) has no atom, then \(\xi\) and X are of the same type.