Most functions are not well-behaved

Statement

This is a meta-principle of calculus, with different forms.

Most functions are not well-behaved

For any definition of "well-behaved", most functions on the reals, or on multiple real variables (or even on intervals within $\mathbb {R}$ or $\mathbb {R} ^{n}$) that can take arbitrary real values do not satisfy that definition.

In particular:

• Most functions do not have explicit expressions.
• Most functions are not continuous.
• Most functions are not integrable.

Two non-identical notions of well-behaved tend to be vastly different

If we have two function properties $p$ and $q$ that both capture some notion of "well-behaved", and there are functions satisfying $p$ but not $q$, then most functions satisfying $p$ do not satisfy $q$.

For instance:

• Most continuous functions are not differentiable (since continuous not implies differentiable).
• Most differentiable functions are not continuously differentiable (since derivative of differentiable function need not be continuous).
• Most integrable functions are not continuous.
• Most infinitely differentiable functions are not analytic.
• Most analytic functions are not polynomials.

In most cases, the world of well-behaved functions is "closed" and the world of non-well-behaved functions is open

Well-behaved functions tend to be closed under various kinds of combination and composition. For instance, continuous functions are closed under addition, subtraction, multiplication, division (except for division by zero issues), and composition. In higher mathematics, we often study the vector space structure, ring structure, and near-ring structure of these well-behaved functions.

For non-well-behaved functions, in contrast, it's generally true that combining a non-well-behaved function and a well-behaved function tends to give another non-well-behaved function. This is most true of addition. It is sometimes less true of multiplication and composition (essentially due to potentially destructive effects on bad behaviors of irreversible operations like multiplication by zero).

Related real-world observation

Most functions constructed from real-world observation, such as measurement of a physical quantity like temperature over time, or measurement of a biological indicator, will tend to be only moderately well-behaved. Although continuity (or something close) is common, explicit expressions are rare, and infinite differentiability is also rare.

Counter-statement

Even though most functions are not well-behaved, it may in many cases be fairly hard to find examples of functions that do not have a particular good behavior. What this statement basically says is that once we find one such example, it is relatively easy to vary it through various operations of combination and composition and obtain several other examples.

Also, although most functions are not well-behaved, most functions we work with on a day-to-day basis are well-behaved. This is a deliberate decision to work in what is essentially a "safe space" of well-behaved function rather than the wild reality.