Every real interval is connected. Otherwise a separation into two nonempty relatively open sets defines a continuous real function equal to zero on one side and one on the other. Choose one point of each set. The intermediate value theorem on the closed segment between them requires the value , a contradiction. The converse also holds: a connected subset of the real line contains every point between two of its points, since omitting an intermediate point separates it into its intersections with the two open half-lines.

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