Is there any statistcal testing method that test whether a given set of data is continous?

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I have some data which I hypothesis that is generated by continuous process. I expect that if I can have high resolution the data will change smoothly (with respect to time). The data I have is for certain intervals and I have no way to zoom in. How can I test my hypothesis that the data were actually continuously generated?

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You can't. Continuity is a modelling assumption based upon your understanding of the process you're investigating.