# What are the properties of functions that cannot be expressed in closed form?

Do they necessarily have asymptotes? Can they be finite over the first interval ($0$ to $x$), infinite over the second ($x$ to $y$), and return to be finite over a third ($y$ to $z$)? When expressed as an infinite series, do the endpoints of their radii of convergence correspond to asymptotes? Thanks.

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It is unclear which functions you are considering. The study of function spaces of various kinds, and the correct definition of a function space to accommodate mathematical concepts (e.g. fourier transform) is a huge area. Bear in mind that in a technical sense "most" continuous functions are not differentiable, and have properties which are not possessed by familiar elementary functions and which seem weird at first glance ... – Mark Bennet Mar 26 '12 at 7:17

To illustrate what I mean, consider these two functions: $$f(x)=\sum_{k=0}^\infty\frac{x^k}{k!},\ \ \ g(x)=\sum_{k=0}^\infty\frac{x^k}{(5k+1)!}.$$ The first one is the exponential, a very respected function with its own notation, i.e. $f(x)=e^x$. The second function is "exotic" (I guess, I didn't make an effort to think about it, the point is just that it isn't one of the canonical functions) but it is probably almost as "good" as the exponential.