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The Hohenberg-Kohn theorem is extended to fractional electron number N, for an isolated open system described by a statistical mixture. The curve of lowest average energy E N versus N is found to be a series of straight line segments with slope discontinuities at integral N. As N increases through an integer M, the chemical potential and the highest occupied Kohn-Sham orbital energy both jump from E M− E M− 1 to E M+ 1− E M. The exchange-correlation potential δ E xc δ n (r→) jumps by the same constant, and lim r→∞ δ E xc δ n (r→)>~ 0.