Derivative of functions with integer range?

[Anonymous No.14522744]

Is there a way to take the derivative of a function that only has an integer range?

Meaning a function that only inputs and outputs integers.

[Anonymous No.14522753]

Floor(f(x))
Floor(df/dx)

[Anonymous No.14522783]

Discrete derivative. Its inverse, the discrete integral is quite useful in deriving closed expressions for summation formulas.

[Anonymous No.14522787]

>>14522783
where can I read about this.

[Anonymous No.14522800]

>>14522787
For a short overview here:
http://homepages.math.uic.edu/~kauffman/DCalc.pdf

[Anonymous No.14522880]

>>14522744
f(x+1) - f(x)

[Anonymous No.14523071]

>>14522744
Yeah. It's just the differences. Same thing with the integral just being the summation.

[Anonymous No.14523230]

You can make an analytical continuation at infinity, and assuming there are no poles, find the derivative everywhere (even in the complex plane).

[Anonymous No.14525006]

>>14522753
why floor?

[Anonymous No.14525015]

>>14522744
[eqn]f'(x) = f(x+1)-f(x)[/eqn]

[Anonymous No.14525229]

>>14522744
Interpolate the function with a spline.

[Anonymous No.14525234]

>>14525015
[math]f(n) = n^2[/math]
[math]f'(n) = (n+1)^2 - n^2 = 2n + 1 [/math]

[Anonymous No.14525378]

>>14525015
>>14525234
How does this work for weirder functions? Pieceiwise, modulus, etc.

[Anonymous No.14525431]

>>14525378
nta but I believe there aren't specific rules you can follow like in the continuous functions. Discrete stuff are usually harder and don't have a specific set of rules. I'll suggest something different though;

You can just find a continuous function passing through the points in the integer ranger you want and then find the derivative in that functions.
Obviously that won't be the discrete derivative that you are asking but you can figure out stuff about the maximums etc. in the integer points.

i.e Given a function [math]f:\mathbb{Z}\to\mathbb{Z}[/math] find a function [math]g:\mathbb{R}\to\mathbb{R}[/math] such that [math]g(n) = f(n)\hspace{0.3cm}(\forall n\in D_f)[/math] (it usually isn't too hard). Then you can can conclude a couple of stuff like if [math]g[/math] is increasing in [math][l,r][/math] then [math]f[/math] is increasing in [math][\left\lceil l \right\rceil, \left\lfloor r \right \rfloor][/math].

[Anonymous No.14525453]

>>14522744
No, and it doesn't even make sense. Most people use differences or something along that line, but think about it physically, how do you define the tangent line at a discreet point? There are an infinite number of tangent lines.

[Anonymous No.14526658]

>>14525453
> There are an infinite number of tangent lines.
> there are an infinite number of tangent lines to 2 distinct points

[Anonymous No.14526685]

>>14522744
yep, and you can have differential equations with them too, know as difference equations.
[eqn]f[n]=f[n-1]+f[n-2] \\
x^2 = x + 1 \\
x^2 - x - 1 = 0 \\
x = \phi [/eqn]
neat, huh?

[Anonymous No.14526687]

>>14525453
>There are an infinite number of tangent lines.
what did he mean by this

[Anonymous No.14528613]

>>14522753
so it only outputs integers?

[Anonymous No.14528652]

>>14522880
/thread
Every other poster ITT is retarded.

[Anonymous No.14530590]

>>14522744
Yeah, given an x, express the change of f(x) at x+1