In this paper simple proofs are given for several propositions about continuity of singular integral operators with Cauchy kernel. Some of these propositions turn out to be consequences of more general tests for continuity of operators of the form $$ (A^hf)(t)\overset{\operatorname{del}}=\int^b_aa(s,t)h(s,t)f(s)ds\quad (t\in[a,b]) $$ under the condition that $A^1$ is a continuous operator (in a given pair of spaces). As the functions $a$ and $h$ one considers, as a rule, functions of the form $1/(e^{it}-e^{is})$ and $\Phi\biggl(\dfrac{\omega(t)-\omega(s)}{e^{it}-e^{is}}\biggr)$ respectively.