Before revision:  And then, we establish the elegant relationships among subdifferentiability, Gateaux-differentiability and Frechet-differentiability for a proper L ^{0 function defined on random normed modules.
After revision:  And then, we establish the elegant relationships among subdifferentiability, G\^ateaux--differentiability and Fr\'ech\'et--differentiability for a proper L ^{0 function defined on random normed modules.