I am trying to create a prediction framework of what a customer will buy on a website, so I am confused if time-connection variable (time of the connection of a customer) is a continuous or discrete variable.
It can be viewed both as discrete and as continuous. In fact we have discrete-time and continous-time models. It depends how did you record the time, e.g. if you count days, or record hours rounded to the nearest hour then it is rather discrete; when you record days, hours and minutes of something happening, then it is closer to continuous. With real-life data we can never be infinitely precise with our measurements, so one could argue that every measurement is discrete given the finite precision. Finally, models designed for continuous variables can in many cases be used with discrete variables (but using models for discrete data with continuous variables is rather a bad idea).