Profiling data by plotting distributions and analyzing summary statistics is a critical step throughout data analysis. Currently, this process is manual and tedious since analysts must write extra code to examine their data after every transformation. This inefficiency may lead to data scientists profiling their data infrequently, rather than after each transformation, making it easy for them to miss important errors or insights. We propose continuous data profiling as a process that allows analysts to immediately see interactive visual summaries of their data throughout their data analysis to facilitate fast and thorough analysis. Our system, AutoProfiler, presents three ways to support continuous data profiling: it automatically displays data distributions and summary statistics to facilitate data comprehension; it is live, so visualizations are always accessible and update automatically as the data updates; it supports follow up analysis and documentation by authoring code for the user in the notebook. In a user study with 16 participants, we evaluate two versions of our system that integrate different levels of automation: both automatically show data profiles and facilitate code authoring, however, one version updates reactively and the other updates only on demand. We find that both tools facilitate insight discovery with 91% of user-generated insights originating from the tools rather than manual profiling code written by users. Participants found live updates intuitive and felt it helped them verify their transformations while those with on-demand profiles liked the ability to look at past visualizations. We also present a longitudinal case study on how AutoProfiler helped domain scientists find serendipitous insights about their data through automatic, live data profiles. Our results have implications for the design of future tools that offer automated data analysis support.