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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 $$\frac {f(x+h)-f(x)}{h}=g(y+1)-g(y)=\Delta g(y).$$

| | | | | created: 2020-03-23 20:57:53 | modified: 2020-03-23 21:13:45 | by: bookofproofs | references: [1112], [8404], [8405]