In analogy to the derivative $f’(x)=\lim_{h\to 0}\frac {f(x+h)-f(x)}{h},$ we introduce the difference operator as follows:

Definition: Difference Operator

Let \(D\subseteq\mathbb R\) (\(D\) is a subset of real numbers) and let \(f:D\to\mathbb R\) be a function. The difference operator $\Delta f$ is defined by $$\Delta f(x)=f(x-1)-f(x).$$


Definitions: 1
Examples: 2
Proofs: 3 4 5 6 7 8 9
Propositions: 10 11 12
Theorems: 13

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  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
  2. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
  3. Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960