when fails theorem of cauchy integral

when fails theorem of cauchy integral

i am interesting what is a condition when following theorem of cauchy
integral fails let $U$ be an open subset of $C$ which is simply connected,
let $f : U ¨ C$ be a holomorphic function, and let $\!\,\gamma$ be a
rectifiable path in $U$ whose start point is equal to its end point. Then

as i understand ,first condition of failing this statement should be that
function should not be holomorphic or function that is not complex
differentiable in a neighborhood of every point in its domain.also maybe
also $U$ substet if it is not connected,then this theorem may fail,what is
also other conditions?thanks in advance