What is the probability that a continuous random variable takes on a single value?

Theoretically, zero. The finite probability can only exist between two points. However, if we are simulating a continuous distribution with a random number generator, all values are truncated to a finite number of digits depending on the register size. If we consider a uniform random number generator, with a 7 digit accuracy, then matching a single value is 1/10,000,000, pretty close to 0.