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discontinuous
Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.
An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.
A continuous variable is a variable for which all possible representations are valid. A discrete variable is a variable for which only some representations are valid. Discontinuous variables apply to data sets where values recorded during particular periods are missing from the set.
Yes. It is a continuous variable. As used in probability theory, it is an example of a continuous random variable.