Lebesgue's decomposition shows that a probability distribution can be decomposed into* absolute continuous*, *discrete*, and* singular-continuous* parts. But in literature, different terminologies are used, which I want to unreval in what follows:

*Continuous*random variables have*absolute-continuous*probability distributions. Their densities are probability density functions.*Discrete*random variables are described by probabilities. Their densities is termed probability mass function. Hence, the distribution is*singular*.- Distributions that are both, singular and continuous, are often called
*singular*.*Singular-continuous*distributions have a cumulative distribution function (devil's staircase) similar to fractals: the function is continuous, but no derivation exists. It has plateaus like that of discrete distributions, but no jumps. In dimensions greater than unity, singular-continuous distributions may be induced by dimensions of absolute-continuous and discrete nature.