Korn's inequality bounds the L^2-norm of the gradient of a vector field by a constant times the L^2-norm of the symmetric part of the gradient. In elasticity theory the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. Korn's inequality is therefore an important tool to prove coerciveness of the strain energy and an a priori estimate for the solution of elastostatic problems. When discretizing nonlinear elasticity problems by nonconforming methods, a discrete counterpart of Korn's inequality is needed to infer the convergence of the numerical scheme. During the talk I will present a proof of a version of Korn's inequality that holds for piecewise regular vector fields and some related discrete functional analysis results.