Subdifferential Rolle's and mean value inequality theorems

In this note we give a subdifferential mean value inequality for every continuous Gâteaux subdifferentiable function f in a Banach space which only requires a bound for one but not necessarily all of the subgradients of f at every point of its domain. We also give a subdifferential approximate Rolle's theorem stating that if a subdifferentiable function oscillates between -ε and ε on the boundary of the unit ball then there exists a subgradient of the function at an interior point of the ball which has norm less than or equal to 2ε.