What does a random continuous function look like?

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My question may be a little strange, but I'm wondering how random continuous functions work? First, how is it possible to define a random continuous function and how to investigate it. For example, is such a function differentiable? Or how many preimages does an image point have on average? Or what does a random continuous mapping of a segment into a square look like? How similar is it to a curve filling a square? Sorry if these are all stupid questions, I just have no idea how to research such questions at all.