A cornerstone of clinical medicine is intervening on a continuous exposure, such as titrating the dosage of a pharmaceutical or controlling a laboratory result. In clinical trials, continuous exposures are dichotomized into narrow ranges, excluding large portions of the realistic treatment scenarios. The existing computational methods for estimating the effect of continuous exposure rely on a set of strict assumptions. We introduce new methods that are more robust towards violations of these assumptions. Our methods are based on the key observation that changes of exposure in the clinical setting are often achieved gradually, so effect estimates must be "locally" robust in narrower exposure ranges. We compared our methods with several existing methods on three simulated studies with increasing complexity. We also applied the methods to data from 14 k sepsis patients at M Health Fairview to estimate the effect of antibiotic administration latency on prolonged hospital stay. The proposed methods achieve good performance in all simulation studies. When the assumptions were violated, the proposed methods had estimation errors of one half to one fifth of the state-of-the-art methods. Applying our methods to the sepsis cohort resulted in effect estimates consistent with clinical knowledge.