Lossy trapdoor functions (LTFs) were first introduced by Peikert and Waters (STOC'08). Since their introduction, lossy trapdoor functions have found numerous applications. They can be used as tools to construct important cryptographic primitives such as injective one-way trapdoor functions, chosen-ciphertext-secure public key encryptions, deterministic encryptions, et al. In this paper, we focus on the lossy trapdoor functions in the presence of continuous leakage. We introduce the new notion of updatable lossy trapdoor functions (ULTFs) and give their formal definition and security properties. Based on these, we extend the security model to the LTFs against continuous leakage when the evaluation algorithm is leakage resilient. Under the standard DDH assumption and DCR assumption, respectively, we show two explicit lossy trapdoor functions against continuous leakage in the standard model. In these schemes, using the technology of matrix kernel, the trapdoor can be refreshed at regular intervals and the adversaries can learn unbounded leakage information on the trapdoor along the whole system life. At the same time, we also show the performance of the proposed schemes compared with the known existing continuous leakage resilient lossy trapdoor functions.