The reason we cannot find the standard deviation of X+Y is because the random variables X and Y are not known to be independent. Standard deviations can be calculated for both discrete and continuous random variables.
The reason we cannot find the standard deviation of X+Y is because the random variables X and Y are not known to be independent. In order to find the standard deviation of the sum of two random variables, they must be independent. Independence means that the occurrence of one event does not affect the probability of the other event happening.
In addition, the statement that standard deviations only apply to discrete random variables is incorrect. Standard deviations can be calculated for both discrete and continuous random variables.