While existing causal discovery methods mostly focus on continuous time series, causal discovery for mixed time series encompassing both continuous variables (CVs) and discrete variables (DVs) is a fundamental yet underexplored problem. Together with nonlinearity and high dimensionality, mixed time series pose significant challenges for causal discovery. This study addresses the aforementioned challenges based on the following recognitions: (1) DVs may originate from latent continuous variables (LCVs) and undergo discretization processes due to measurement limitations, storage requirements, and other reasons. (2) LCVs contain fine-grained information and interact with CVs. By leveraging these interactions, the intrinsic continuity of DVs can be recovered. Thereupon, we propose a generic deep mixed time series temporal causal discovery framework. Our key idea is to adaptively recover LCVs from DVs with the guidance of CVs and perform causal discovery in a unified continuous-valued space. Technically, a new contextual adaptive Gaussian kernel embedding technique is developed for latent continuity recovery by adaptively aggregating temporal contextual information of DVs. Accordingly, two interdependent model training stages are devised for learning the latent continuity recovery with self-supervision and causal structure learning with sparsity-induced optimization. Experimentally, extensive empirical evaluations and in-depth investigations validate the superior performance of our framework. Our code and data are available at https://github.com/chunhuiz/MiTCD.