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We study the flavor-changing-neutral-current process c -> u mu(+)mu(-) using 1.3 fb(-1) of p (p) over bar collisions at root s = 1.96 TeV recorded by the D0 detector operating at the Fermilab Tevatron Collider. We see clear indications of the charged-current mediated D(s)(+) and D(+)->phi pi(+)->mu(+)mu(-)pi(+) final states with significance greater than 4 standard deviations above background for the D(+) state. We search for the continuum neutral-current decay of D(+)->pi(+)mu(+)mu(-) in the dimuon invariant mass spectrum away from the phi resonance. We see no evidence of signal above background and set a limit of B(D(+)->pi(+)mu(+)mu(-))< 3.9 x 10(-6) at the 90% C.L. This limit places the most stringent constraint on new phenomena in the c -> u mu(+)mu(-) transition.