How do I know if the given signal is discrete?
Fortunately, one typically does not receive signals from strangers in real life. One knows because one is working on an application. 

A discrete signal can be equivalent to a continuous signal so you need a set of assumptions tying the two together, like a Nyquist sampling criteria and then you need to test for equivalence under those assumptions. 

A function that only exists at discrete points( like time points) and there is no meaningful notion of intermediate values between those points is discrete. Some financial time series are like that.  Arrival times of a Poisson process is another discrete function.  If intermediate values on a continuous range are meaningful, a continuous function is likely.