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Examples of continuous variation are anything that can be measured such as, shoe size, height, weight, hand span and diameter of limpit shells. Discontinuous variation however is when there is a clear cut difference such as different colours or different species.
Sure! The definition of Laplace transform involves the integral of a function, which always makes discontinuous continuous.
Continuous variation is a variation that is distributable; under a normal curve. Height is an example of this with all heights being along a continuum of heights within populations, at least. This distribution of traits is usually controlled by many alleles in a additive fashion. Polygenic. Discontinuous variation is of one trait, allele, or the other. Blood groups are an example of this. A, B , O. You can only have two alleles here, so AA and AB and OO and AO, AB etc. are the expressed ( less the recessive O, except homozygous ) traits. These are single variations based on one allele and are not distributable along a continuum.
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.