Following up on the previous MO question "Are there any important mathematical concepts without discrete analogue?", I'd like to ask the opposite: what are examples of notions in math that were not originally discrete, but have good discrete analogues? While a few examples arose in the answers to that earlier MO question, this wasn't what that question was asking, so I'm sure there are many more examples not mentioned there. What reminded me of this older MO question was seeing an MO question "Why is the Laplacian ubiquitous?", since that is an instance of an important notion which has a discrete analgoue.

In an answer, it would be interesting to hear about the relationship between the continuous and discrete versions of the notion, if possible, and references could also be helpful. Thanks!

I have not made this community wiki, because I thought that writing a good answer could take some effort.