  Existing statistical methods in causal inference often rely on the assumption
that every individual has some chance of receiving any treatment level
regardless of its associated covariates, which is known as the positivity
condition. This assumption could be violated in observational studies with
continuous treatments. In this paper, we present a novel integral estimator of
the causal effects with continuous treatments (i.e., dose-response curves)
without requiring the positivity condition. Our approach involves estimating
the derivative function of the treatment effect on each observed data sample
and integrating it to the treatment level of interest so as to address the bias
resulting from the lack of positivity condition. The validity of our approach
relies on an alternative weaker assumption that can be satisfied by additive
confounding models. We provide a fast and reliable numerical recipe for
computing our estimator in practice and derive its related asymptotic theory.
To conduct valid inference on the dose-response curve and its derivative, we
propose using the nonparametric bootstrap and establish its consistency. The
practical performances of our proposed estimators are validated through
simulation studies and an analysis of the effect of air pollution exposure
(PM$_{2.5}$) on cardiovascular mortality rates.
