Almost-everywhere convergence of wavelet transforms of Lp-functions under minimal conditions on wavelets was proved by Rao et al. in 1994. However, results on convergence almost everywhere do not provide any information regarding the exceptional set (of Lebesgue measure zero), where convergence does not hold. We prove that if a wavelet ψ satisfies a single additional condition xψ(x) ∈ L1 (R), then, instead of almost-everywhere convergence, we have a more sophisticated result, i.e. convergence of wavelet transforms everywhere on the entire Lebesgue set of Lp-functions. For example, wavelets with compact support, used frequently in applications, obviously satisfy this extra condition. Moreover, we prove that our conditions on wavelets ensure the Riemann localization principle in L1 for the wavelet transforms.