Exponential stability of linear systems with time-varying piecewise-continuous delays is studied. It is assumed that the delay function has a form of a sawtooth with a constant delay derivative Τ ≠ 0. In the recent paper (Fridman, 2009) piecewise-continuous (in time) Lyapunov-Krasovskii Functionals (LKFs) have been suggested for the stability analysis of sampled-data systems (with Τ = 1) in the framework of input delay approach. Differently from the existing time-independent LKFs for systems with time-varying delays, the discontinuous ones can guarantee the stability under the sampling which may be greater than the analytical upper bound on the constant delay that preserves the stability. The objective of the present paper is to extend the piecewise-continuous LKF method to systems with a general sawtooth delay. The discontinuous terms of LKFs improve the results for all values of Τ, though the most essential improvement corresponds to Τ = 1.