  Submodular continuous functions are a category of (generally)
non-convex/non-concave functions with a wide spectrum of applications. We
characterize these functions and demonstrate that they can be maximized
efficiently with approximation guarantees. Specifically, i) We introduce the
weak DR property that gives a unified characterization of submodularity for all
set, integer-lattice and continuous functions; ii) for maximizing monotone
DR-submodular continuous functions under general down-closed convex
constraints, we propose a Frank-Wolfe variant with $(1-1/e)$ approximation
guarantee, and sub-linear convergence rate; iii) for maximizing general
non-monotone submodular continuous functions subject to box constraints, we
propose a DoubleGreedy algorithm with $1/3$ approximation guarantee. Submodular
continuous functions naturally find applications in various real-world
settings, including influence and revenue maximization with continuous
assignments, sensor energy management, multi-resolution data summarization,
facility location, etc. Experimental results show that the proposed algorithms
efficiently generate superior solutions compared to baseline algorithms.
