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Lieb-Thirring inequalities give a bound on power sums of the negative eigenvalues of a Schroedinger operator in terms of an L^p norm of the potential. Motivated by a study of the stability-of-matter problem in quantum mechanics, they have become a useful tool in many areas of analysis. We present an extension of these inequalities to analogous inequalities for perturbations of the continuous spectrum of the Laplacian by local potentials. In physics terms, we bound the change in energy of an ideal Fermi gas when the density is changed locally, which is an important quantity in condensed matter physics. Our bounds show that, up to a universal constant, the semiclassical approximation gives the correct answer. (This is joint work with R. Frank, M. Lewin and E. Lieb.)