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In 1977 P. Yang asked whether there exist complete immersed complex submanifolds ϕ: M k → C N with bounded image. A positive answer is known for holomorphic curves (k = 1) and partial answers are known for the case when k > 1. The principal result of the present paper is a construction of a holomorphic function on the open unit ball B N of C N whose real part is unbounded on every path in B N of finite length that ends on bB N . A consequence is the existence of a complete, closed complex hypersurface in B N . This gives a positive answer to Yang's question in all dimensions k, N, 1 ≤ k < N , by providing properly embedded complete complex manifolds.

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