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We define continuous incrementally stationary channels with finite incremental input- and output-memories and prove the coding theorem and its converse for such channels. These channels include, as special cases, stationary channels with finite input- and output-memories and incrementally stationary and memoryless channels. The former is defined here and the latter has been defined previously. It is emphasized that, with an elementary measure-theoretic formulation, the standard method of proving the coding theorem for discrete channels becomes directly applicable for continuous channels. Consequently, the tedious step of representing a continuous channel by an infinite series of discrete channels can be avoided entirely.