# A problem on continuity and differentiability

• Jun 26th 2007, 02:29 AM
ppcparichoy
A problem on continuity and differentiability
Q: Let f:[0,1]->[0,1] be a function defined as follows.
Take x in [0,1] with the decimal expansion x=.x1x2x3x4x5.....
Map it to y in [0,1] such that y=.x1x3x5......
Prove that this map is continuous but nowhere differentiable in [0,1].

I need some help for this problem.

If anybody has a solution please mail me at ppcparichoy@gmail.com
• Jun 26th 2007, 06:42 AM
ThePerfectHacker
This is not even a map.

Consider $\displaystyle x=.1111111.... = .1099999....$
Ones gets "mapped" to,
$\displaystyle .1111111...$
The other is "mapped" to,
$\displaystyle .1999999...$
The results are clearly not the same.
• Jun 26th 2007, 09:16 PM
ppcparichoy
No this is not correct
x=0.111111.... is not equal to 0.1099999.....

Clearly 1st one is 1/9 and the second one is 11/100.
• Jun 26th 2007, 11:06 PM
CaptainBlack
Quote:

Originally Posted by ppcparichoy
x=0.111111.... is not equal to 0.1099999.....

Clearly 1st one is 1/9 and the second one is 11/100.

But the same objection applies to x1=0.1100000.. and x2=0.109999.. which
are equal, but one being mapped to 0.10000.. and the other to 0.19999..=0.2000..,
which are not equal.

RonL
• Jun 27th 2007, 05:02 AM
topsquark
Quote:

Originally Posted by ppcparichoy
Q: Let f:[0,1]->[0,1] be a function defined as follows.
Take x in [0,1] with the decimal expansion x=.x1x2x3x4x5.....
Map it to y in [0,1] such that y=.x1x3x5......
Prove that this map is continuous but nowhere differentiable in [0,1].

I need some help for this problem.

If anybody has a solution please mail me at ppcparichoy@gmail.com

I think what he is trying to say is that we have a map such that the decimal 0.x1x2x3x4x5..... maps to 0.x1x3x5......

For example if x = 1 then
0.1112131415... maps to 0.11315....

If x = 2 then
0.2122232425... maps to 0.212325...

-Dan
• Jun 27th 2007, 10:42 AM
CaptainBlack
Quote:

Originally Posted by topsquark
I think what he is trying to say is that we have a map such that the decimal 0.x1x2x3x4x5..... maps to 0.x1x3x5......

For example if x = 1 then
0.1112131415... maps to 0.11315....

If x = 2 then
0.2122232425... maps to 0.212325...

-Dan

Problematic as $\displaystyle x \in [0,1],\ x=0.x1x2x3...$, the interpretation:

$\displaystyle x \in [0,1],\ x=0.x_1x_2x_3...$, with $\displaystyle x_i \in \{0,1,2, .. ,9\}$

seems more natural, but it looks ambiguous to me any way.

RonL