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We study the propagation of hard x rays in single curved x-ray waveguide channels and observe waveguide effects down to surprisingly small radii of curvature R≃10 mm and a large contour length s≃5 mm, deflecting beams up to 30°. At these high angles, about 2 orders of magnitude above the critical angle of total reflection θ(c), most radiation modes are lost by "leaking" into the cladding, while certain "survivor" modes persist. This may open up a new form of integrated x-ray optics "on a chip," requiring curvatures mostly well below the extreme values studied here, e.g., to split and to delay x-ray pulses.