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  "body": "\n## Summary\n\nThe Lipschitz class $\\mathrm{Lip}\\,\\alpha$, for $0 \u003c \\alpha \\le 1$, is the set of functions on an interval that satisfy $\\abs{f(x) - f(y)} \\le M \\abs{x - y}^\\alpha$ for some constant $M$.\nFor $\\alpha = 1$ these are the [[Lipschitz continuous|0QNVQ4VD9T]] functions, and for $\\alpha \u003c 1$ the [[Hölder continuous|012PBA0FMD]] functions of exponent $\\alpha$.\nIn approximation theory these classes are the standard scale of smoothness.\nFor $0 \u003c \\alpha \u003c 1$, a continuous $2\\pi$-periodic function is in $\\mathrm{Lip}\\,\\alpha$ exactly when its best trigonometric approximation of degree $n$ has error of order $n^{-\\alpha}$.\n\n## Explanations\n\n### L1 human\n\nThe exponent $\\alpha$ says how fast the function may change over short distances.\nWith $\\alpha = 1$ the change is at most proportional to the distance, like a graph with bounded slope.\nWith smaller $\\alpha$ the function may be steeper near some points: $\\sqrt{x}$ on $[0, 1]$ is in $\\mathrm{Lip}\\,\\tfrac12$ but not in $\\mathrm{Lip}\\,1$, because its slope blows up at $0$.\nSmaller $\\alpha$ means a larger class of rougher functions.\n",
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