A complex function is said to be analytic on a region $R$ if it is complex differentiable at every point in $R$.
However, I read that if a function only differentiable on a straight line on the complex plane it is not analytic on any domain. Does this mean "region $R$" should be understood as open set, instead of any set? That is, when referring to analytic complex function, it is only with respect to an open set?