# Is there a class of functions that have a cyclical or constant derivative?

I'm working on approximating functions in A.I., and I noticed that everyday functions seem to, at some point, have either a cyclical, or constant, derivative.

For example, a straight line has a constant derivative, and so we can easily approximate a linear function by simply taking the difference between the range values.

Consider f(x) = 2x + 3:

I would start with a dataset that contains the range values, and then try to extrapolate a function, so I would be given (x,y) pairs -

(1,5); (2,7); (3,9); (4,11).

If we take the difference over the y values, it's 2, which is constant, and of course the slope of the line. This means that we can generate the function by simply taking the initial y value 5, and iteratively adding 2.

If you do the same for a sin function, eventually you'll find the derivative repeats.

If we consider a parabola, the function itself does not have a constant rate of change, since its slope is a linear function. But, this of course implies that its derivative will have a constant rate of change.

As a general matter, I'm wondering if there's a word that describes this class of functions that eventually have a constant, or cyclical derivative.

Note that it is necessarily the case that this class of functions can be very easily approximated, since you just bootstrap up from the first constant / cyclical derivative, until you get to the original function.

As a result, it looks like this is a pretty good method for approximating normal data that doesn't have a wildly complex underlying function.

• Isn't this $\{f:f\text{ is linear}\}\cup\{g:g\text{ is periodic and differentiable}\}$? – 79037662 Oct 31 at 21:13
• Sounds like $\{f : f'$ is periodic $\}$ to me. If you integrate a periodic function over its period and obtain 0, then the integral will be another periodic function. This makes me think the set we're looking for is a subset of periodic functions, shifted down to satisfy the integral constraint. – dskeletov Oct 31 at 21:18
• I don't think that's right, but apologies if I'm misunderstanding you. For example, the integral of a linear function is never 0, but a linear function has the property I'm referring to. – Feynmanfan85 Oct 31 at 21:19
• Polynomials, and only polynomials, eventually have constant derivative. Periodic functions have periodic derivatives. – Gerry Myerson Oct 31 at 22:01
• That seems sensible. Is there a class of function that eventually has a periodic derivative, but is not itself periodic? – Feynmanfan85 Nov 1 at 1:40