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  "body": "\n## Definition\n\nA *time scale* is a nonempty [[closed subset|0ZCK8DQYN1]] $\\mathbb{T}$ of $\\R$.\nThe forward jump operator $\\sigma \\colon \\mathbb{T} \\to \\mathbb{T}$ is $\\sigma(t) = \\inf \\set{s \\in \\mathbb{T} : s \u003e t}$, with $\\sigma(t) = t$ when $t$ is the largest element of $\\mathbb{T}$.\nThe graininess is $\\mu(t) = \\sigma(t) - t$.\nLet $\\mathbb{T}^\\kappa$ be $\\mathbb{T}$ without its largest element $m$ if $m$ exists and $\\sup \\set{s \\in \\mathbb{T} : s \u003c m} \u003c m$; otherwise $\\mathbb{T}^\\kappa = \\mathbb{T}$.\nLet $f \\colon \\mathbb{T} \\to \\R$ and $t \\in \\mathbb{T}^\\kappa$.\nThe function $f$ is *delta differentiable* at $t$, with *delta derivative* $f^\\Delta(t) \\in \\R$, if for every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that\n$$\\abs{f(\\sigma(t)) - f(s) - f^\\Delta(t) \\, (\\sigma(t) - s)} \\le \\varepsilon \\abs{\\sigma(t) - s} \\quad \\text{for all } s \\in \\mathbb{T} \\text{ with } \\abs{t - s} \u003c \\delta.$$\n*Time-scale calculus* is the study of delta derivatives, of the matching integrals, and of dynamic equations built from them.\n\n## Remarks\n\nThe delta derivative is unique when it exists.\nExample: for $\\mathbb{T} = \\R$, $\\sigma(t) = t$ and the condition says that $f$ is differentiable at $t$ with $f^\\Delta(t) = f'(t)$, the ordinary [[derivative|0V1EJCT8G6]].\nExample: for $\\mathbb{T} = \\Z$, $\\sigma(t) = t + 1$, and with $\\delta = \\frac{1}{2}$ the only admissible $s$ is $t$, so the condition holds exactly for $f^\\Delta(t) = f(t + 1) - f(t)$.\nMore generally, if $\\mu(t) \u003e 0$ and $f$ is continuous at $t$, then $f^\\Delta(t) = \\frac{f(\\sigma(t)) - f(t)}{\\mu(t)}$, a [[difference quotient|AZBDB5F4M4]].\nNon-example: on $\\mathbb{T} = \\R$, the function $f(t) = \\abs{t}$ is not delta differentiable at $0$, because it has no ordinary derivative there.\n",
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