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I think it's useless to try and evaluate the order of a function where the parameter doesn't have a strict static type, because whatever order of function you have, you can always make it have an order one higher. So basically,

In your example, f is simply a higher-order function. That's it. In a typed programming language it returns a function of the same order as its argument, and the order of the function itself is the order of the argument plus one.

So a 0th order function is a constant?

I think it ends up trivially rank 1 because it's a degenerate case, and though it might not make a difference here, there are different type systems that would give different types for things, using different constructs to extend plain old Cartesian closed categories e.g. subtypes vs. Type classes.

I noticed because of the font and icons. I use PyCharm

Fooooobbbaarrrr

[humorous reply]

And together with currying you can get something like functions with an order of n.

My all time favourite foot-gun lol I wish my IDE had an extension that tracked the frequency and types of exceptions so that I could chart it from time to time. That'd be a lot of fun to track over time.

By the way, I do fun things to TeX macros.

Is it because inline isn’t dynamic?