**Contents:**

## What is a Smooth Function?

A **smooth function** is just like the name sounds: it’s a function that travels without any drop offs, jumps or other strange behavior that would make it not differentiable. More specifically, the function is differentiable up to some desired point.

That desired point is called the “*class*“. It is denoted:

**C ^{n}**

Where “n” is the order.

An **order** is just the number of derivatives. For example, a first derivative is order 1, a fourth derivative is order 2, and a function that can be differentiated an infinite number of times is order ∞.

Smooth functions include exponential functions, trigonometric functions and many others.

## Smooth Function on an Open Interval

A real-valued, smooth function of class C^{n}, defined on an open interval has the following characteristics:

- The
**class**(*n*) is in the set of natural numbers; this is written as n ∈ ℕ (Natural numbers are counting numbers, i.e. whole, non-negative numbers). - The function is defined on the
**open interval**(a, b). An “open interval” doesn’t contain the endpoints. **All derivatives exist up to order n**, on the stated interval. For example, a smooth function of class C^{2}has both a first derivative and a second derivative. If all derivatives exist, the function is called*infinitely smooth*or*infinitely differentiable*.- The derivatives are
**continuous**. In other words, its derivative is a continuous function.

## Smooth Function on a Closed Interval [a, b]

Smooth functions can also be defined on a closed interval [a, b]. The definitions are the same as that for open intervals, except that the **closed interval includes endpoints**. The function ends abruptly at the endpoints of a closed interval, so the function can’t be described as “smooth” at that exact point without additional information about what happens at “a” and “b”.

Specifically, a smooth function of class C^{n}, defined on [a, b], has continuous, one-sided derivatives at point a and b.

- For point a, the one-sided derivative from the right exists.
- For point b, the one-sided derivative from the left exists.

These derivatives must be the same order (i.e. both must be differentiable the same x number of times).

Note that if the function is **half closed**: (a, b] or [a, b), then only the closed side will have a one-sided derivative.

## What is a Flat Function?

Informally speaking, a **flat function** has a slope of zero; They are flat, in the same sense that a floor of a house is flat. A more strict definition is that it is a smooth function with derivatives that vanish to zero at a given point. In other words, the values for the derivative get smaller and the slope flattens as it approaches that point.

An **infinitely flat function** will have all its derivatives (i.e. an infinite number of them) equal zero. For practical purposes (i.e. having to test all derivatives up to the ∞^{th}), flat functions are defined for being flat to a certain *degree*. A “third degree flat function” for example, has all the derivatives up to the third, checked as being equal to zero.

All constant functions are flat functions, so do not need to have their derivatives checked. However, for other functions you can’t assume true flatness unless you can prove that every possible derivative is flat. Otherwise, you should note the degree of flatness.

For example, the even function f(x) = x^{2} is flat at x = 0 (at least to the tenth degree, which is as far as I checked):

Flat functions are important in real analysis because they do not have a meaningful Taylor series expansion around 0; the non-constant part of the function always lies in the series remainder [1].

## Flat Function Examples

The following well-known example of a flat function is flat at the origin and not analytic at any point:

As this particular function is known to have all-zero derivatives at the origin, it is sometimes called *The* flat function.

Other examples [2]:

- f(x) = exp(-1/x) on (0, 1] and f(0) = 0,
- on [0. 1]

## Maximally Flat Function

A **maximally flat function** has as many of its derivatives as possible equal to zero [3]. Another way to put this: we want the function to be flat at a certain point and to change from “flatness” as slowly as possible.

Given a function fitting a certain form, the goal is to fit the “best” function to the formula— the one with the maximum number of zero derivatives (first, second,…n^{th} derivative). The following image shows a few functions that fit the form f(x) = ax^{3} + bx^{2} + cx. The maximally flat function at x = 5 is the function f(x) = 5x^{3} – 15x^{2} + 15x [4], because it’s the one with the slope that “steepens” the slowest as we move away from f(x) = 5:

## References

[1] Pan, Y. & Wang, M. When is a Function not Flat? Retrieved May 7, 2021 from: http://www.stat.uchicago.edu/~meiwang/research/continuation.pdf

[2] Stoica, G. When Must a Flat Function be Identically 0? Retrieved May 7, 2021 from: https://www.tandfonline.com/doi/abs/10.1080/00029890.2018.1470415?journalCode=uamm20

[3] Ghausi, M. (1971). Electronic Circuits Devices, Models, Functions, Analysis, and Design. The University of Michigan.

[4] Maximally Flat Transformer Functions.doc. (2020). Retrieved May 7, 2021 from: http://www.ittc.ku.edu/~jstiles/723/handouts/Maximally%20Flat%20Transformer%20Functions.pdf

Shikin, E. (2014). Handbook and Atlas of Curves. CRC Press.