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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 17 "Devil's Staircase" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 471 "The Devil's Stair
case is the graph of the continuous Cantor function, which has derivat
ive = 0 on the complement of the Cantor set, i.e. on 100% of the inter
val [0,1] . Yet, this function is NOT constant.  You may have learned \+
in Calculus (with the Mean Value Theorem) that if the derivative of a \+
function is zero EVERYWHERE on an interval, then the function is const
ant on that interval.  In fact this is consistent with the fact that t
he Devil's staircase is horizontal" }}{PARA 0 "" 0 "" {TEXT -1 331 "on
 the intervals that were thrown out to make the Cantor set.  The Canto
r function manages to do all of its growing on the Cantor set, which h
as 0% of the interval length, but still has a lot of points.  (And the
 derivative of the Cantor function is +infinity at every point in the \+
Cantor set, which makes for very steep climbing.)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "g1:=P->AFFI
NE1(P,1/3.,0,0,1/2.,0,0);\n      #scale by 1/3 in x-dir, by 1/2 in y-d
ir,\n      #and don't translate \ng2:=P->AFFINE1(P,1/3.,0,0,0,1/3.,.5)
;\n      #squash out the y-direction, scale x dir by .5,\n      #and t
ranslate by [1/3.,.5]\ng3:=P->AFFINE1(P,1/3.,0,0,1/2.,2/3.,1/2.);\n   \+
   #scale as in g1, then translate by [2/3,1/2]" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 112 "TESTMAP([g1,g2,g3]);  #the whole L-box got sq
uashed into a line\n    #segment by g2, that's why you don't see it." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "S:=\{[0,0]\}:#initial se
t consisting of one point\nfor i from 1 to 9 do\nS1:=map(g1,S);\nS2:=m
ap(g2,S);\nS3:=map(g3,S);\nS:=`union`(S1,S2,S3);\nod:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "pointplot(S,symbol=point,scaling=co
nstrained,\n    title=`Cantor's devil's staircase`);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "11" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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