I presume that it is because one needs to take a limit to define the derivative, but is there another reason?
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You may want to take a look at When is a Function that Satisfies the Cauchy-Riemann Equations Analytic? by J. D. Gray and S. A. Morris, The American Mathematical Monthly, Vol. 85, No. 4, (Apr., 1978), pp. 246-256.
Yes, in a sense, we want open domains so limits can be computed, but we have a notion of directional limit, so this is not much of a reason; what I mean is, some justification is needed that we actually lose something otherwise.
The paper above gives examples showing how the Cauchy-Riemann equations alone do not suffice for analyticity, even in the presence of continuity, and proceeds to look for weak additions to these two requirements.