"Entropy" roughly captures the degree of "information" in a probability distribution.
For discrete distributions there is a far more exact interpretation: The entropy of a discrete random variable is a lower bound on the expected number of bits required to transfer the result of the random variable.
But for a continuous random variable, there are uncountably infinite number of outcomes, so we cannot even begin to transfer which exact outcome has occurred in a finite string of bits.
What is an equivalent interpretation of entropy for continuous variables?