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 $$\Delta f(x)=f(x-1)-f(x).$$

### Notes

• $D$ is assumed to contain both, $x$ and $x+1.$
• Unlike the derivative $f'(x)$, the difference operator $\Delta f(x)$ always exists, provided, $f(x)$ is defined.
• Occasionally, we can reduce a given differential operator $\frac {f(x+h)-f(x)}{h}$ by replacing $x$ by the normalized variable $y:=\frac xh$ and $f(x)$ by the normalized function $g(x/h):= f(x)/h.$ Then $$\frac {f(x+h)-f(x)}{h}=g(y+1)-g(y)=\Delta g(y).$$

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

Github: ### References

#### Bibliography

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