Derivitive and Integral of Flooring and Ceiling?

[MWresearch]

If you have a flooring or ceiling function somewhere, how do you differentiate it and integrate it?

[Bignose]

well, there are obvious discontinuities in the function, so you have to account for that. It doesn't let you use the standard definition of the derivative. The integral can be done in parts.

[MWresearch]

Well it doesn't necessarily have to be those isolated functions, I was thinking more for the chain rule, like if the flooring function was nested inside something else that ultimately produced a continuous curve.

[MWresearch]

So...it doesn't have a derivative while nested inside another function that produces a continuous curve? Or....

Do I reuse it in such a way that it is its own derivative or integral? Like d(floor(x)^2)/dx = 2(floor(x))?

[Bignose]

No. If you think about it, floor(x) is a constant between the integers. It's derivative would be 0 there and undefined at its discontinuities.

[MWresearch]

I just find it very odd that I would have to break a completely continuous, smooth, monotonic curve into these separate parts that aren't even graphically visible at the points of the derivative or ceiling function.

If I had y=x^2, but replaced x with "floor(x)," well the lowest integer between 1 and 1 is 1, and the lowest integer between 2 and 2 is 2 and so on and it should be a continuous curve, yet I still have to break down the simple integral of x^2 into parts at locations that I can't even directly see?

Edited March 8, 2015 by MWresearch

[John]

The function y = (floor(x))^2 isn't continuous. See the graph at WolframAlpha: http://www.wolframalpha.com/input/?i=y+%3D+floor%28x%29%5E2

As Bignose mentioned, the derivative of the floor function is 0 for all non-integer x and undefined for all integer x. Thus, using the chain rule, we see that the derivative of floor(x)^2 = 2floor(x) * 0 = 0 for non-integer x, and of course it's undefined for integer x.

Also, as a minor nitpick, y = x^2 isn't monotonic.

Edit: My second paragraph is worded a bit poorly, as it seems to imply that non-differentiability implies discontinuity, which is certainly false (in fact, most continuous functions are nowhere differentiable). But since the derivative of floor(x)^2 was mentioned earlier, I'll leave it in anyway, for its general point.

Edited March 9, 2015 by John

[studiot]

This should not be an argument about continuity.

 

The calculus you require for the floor and ceiling functions is called the finite calculus.

 

In this calculus the derivative operator D is replaced by the difference operator [math]\Delta [/math]

and the integral operator [math]\int {} [/math]is replaced by the Summation operator [math]\Sigma [/math]

 

You can find out about these and the maths of floor and ceiling function by reading

 

Graham, Knuth and Patashank

 

Section 2.6 deals with the finite calculus and chapter 3 with f&c functions.

Edited March 9, 2015 by studiot

[MWresearch]

Oh, that is my bad then. Somehow I got continuous curves involving the flooring function when I was looking at weird integrals for functions involving the lambert w function and the gamma function and so I assumed that the "picking between numbers" operation applied equally to decimals.

I also just meant monotonic because I was only looking at the first quadrant of x^2.