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Derivitive and Integral of Flooring and Ceiling?


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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))?

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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 by MWresearch
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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 by John
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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 by studiot
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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.

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