Thread: 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

This is not even a map.

Consider
Ones gets "mapped" to,

The other is "mapped" to,

The results are clearly not the same.

x=0.111111.... is not equal to 0.1099999.....

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

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

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

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 , the interpretation:

, with

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

RonL