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To creat the function ", StyleBox["f[x]", "Input"], " in ", StyleBox["Mathematica", FontSlant->"Italic"], " we should use the built-in function ", StyleBox["RealDigits[x]", "Input"], "which returns a list of (non-zero) digits in ", StyleBox["x", "Input"], " and the appropriate exponent to give the correct number of initial \ zeros." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RealDigits[ .0123]\)], "Input"], Cell[OutputFormData["\<\ {{1, 2, 3}, -1}\ \>", "\<\ {{1, 2, 3}, -1}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "If ", StyleBox["x = .abc... = a 10^-1 + b^10^-2 + c^-3 + ...", "Input"], ", then\n", StyleBox["f[x] = a 10^-2 + b 10^-4 + c 10^-6 + ...", "Input"], " We use double brackets to pick out the pieces of the lists returned by ", StyleBox["RealDigits[]", "Input"], ". The list of digits is stored in ", StyleBox["d", "Input"], " and the exponent in ", StyleBox["e", "Input"], ". We then compute ", StyleBox["f[x]", "Input"], " as a sum of the digits in ", StyleBox["d", "Input"], " mutiplied by ", StyleBox["10", "Input"], " raised to doubled exponents:" }], "Text"], Cell[BoxData[ \(f[x_]\ := \ \((r = \ \ RealDigits[x]; \ d\ = \ r[\([1]\)]; e\ = r[\([2]\)]; Sum[\ d[\([n]\)]*10.0^\((2 \((e - n)\))\), \ {n, 1, Length[d]}]) \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f[ .0123]\)], "Input"], Cell[OutputFormData["\<\ 0.0001020299999999999\ \>", "\<\ 0.00010203\ \>"], "Output"] }, Open ]], Cell["The plot looks like a step function:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p\ = \ Plot[f[x], \ {x, 0, 1}, \n\t\t\ \ \ \ PlotRange -> {0, 0.1}, PlotPoints -> 100, \ AspectRatio -> 1]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier 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