130 3 Real Analysis
The functionφis extended linearly over each open interval that was removed in the
process of constructingC, to obtain a continuous surjectionφ:[ 0 , 1 ]→[ 0 , 1 ]×[ 0 , 1 ].
This concludes the proof of the theorem.
To visualize this Peano curve, consider the “truncations’’ of the Cantor setC 1 ={
0 ,
1
3
,
2
3
, 1
}
,C 2 =
{
0 ,
1
9
,
2
9
,
1
3
,
2
3
,
7
9
,
8
9
, 1
}
,
C 3 =
{
0 ,
1
27
,
2
27
,
1
9
,
2
9
,
7
27
,
8
27
,
1
3
,
2
3
,
19
27
,
20
27
,
7
9
,
8
9
,
25
27
,
26
27
, 1
}
,
C 4 =
{
0 ,
1
81
,
2
81
,
1
27
,
2
27
,
7
81
,
8
81
,
1
9
,
2
9
,
19
81
,
20
81
,
7
27
,
8
27
,
25
81
,
26
81
,
1
3
,
2
3
,
55
81
,
56
81
,
19
27
,
20
27
,
61
81
,
62
81
,
7
9
,
8
9
,
73
81
,
74
81
,
25
27
,
26
27
,
79
81
,
80
81
, 1
}
,...,
and defineφn:Cn→[ 0 , 1 ]×[ 0 , 1 ],n≥1, as above, and then extend linearly. This
gives rise to the curves from Figure 20. The curveφis their limit. It is a fractal: if we
cut the unit square into four equal squares, the curve restricted to each of these squares
resembles the original curve.
n= 12 n= n= 3 n=^4
Figure 20386.Letf:R→Rbe a continuous function satisfyingf(x)=f(x^2 )for allx∈R.
Prove thatfis constant.
387.Does there exist a continuous functionf:[ 0 , 1 ]→Rthat assumes every element
of its range an even (finite) number of times?
388.Letf(x)be a continuous function defined on[ 0 , 1 ]such that
(i)f( 0 )=f( 1 )=0;
(ii) 2f(x)+f(y)= 3 f(^2 x 3 +y)for allx, y∈[ 0 , 1 ].
Prove thatf(x)=0 for allx∈[ 0 , 1 ].
389.Letf:R→Rbe a continuous function with the property that
lim
h→ 0 +f(x+ 2 h)−f(x+h)
h= 0 , for allx∈R.Prove thatfis constant.