Prove that the derivative of the Cantor Function is zero almost everywhere.
$begingroup$
Construct a continuous increasing function $f:[0,1] to [0,1]$ such that $f(0) = 0$, $f(1)=1$ and $f'(x) = 0$ in a open dense set.
The Cantor Function works. I know that the derivative of Cantor Function is zero almost everywhere, but I cannot prove it. Can someone help me?
real-analysis metric-spaces
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
$endgroup$
add a comment |
$begingroup$
Construct a continuous increasing function $f:[0,1] to [0,1]$ such that $f(0) = 0$, $f(1)=1$ and $f'(x) = 0$ in a open dense set.
The Cantor Function works. I know that the derivative of Cantor Function is zero almost everywhere, but I cannot prove it. Can someone help me?
real-analysis metric-spaces
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
$endgroup$
6
$begingroup$
Check that the Cantor function is constant on each connected component of the complement of the Cantor set. This should be essentially immediate by construction.
$endgroup$
– Andrés E. Caicedo
Jan 3 at 20:34
add a comment |
$begingroup$
Construct a continuous increasing function $f:[0,1] to [0,1]$ such that $f(0) = 0$, $f(1)=1$ and $f'(x) = 0$ in a open dense set.
The Cantor Function works. I know that the derivative of Cantor Function is zero almost everywhere, but I cannot prove it. Can someone help me?
real-analysis metric-spaces
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
$endgroup$
Construct a continuous increasing function $f:[0,1] to [0,1]$ such that $f(0) = 0$, $f(1)=1$ and $f'(x) = 0$ in a open dense set.
The Cantor Function works. I know that the derivative of Cantor Function is zero almost everywhere, but I cannot prove it. Can someone help me?
real-analysis metric-spaces
real-analysis metric-spaces
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
asked Jan 3 at 20:31
Lucas CorrêaLucas Corrêa
1,5581421
1,5581421
6
$begingroup$
Check that the Cantor function is constant on each connected component of the complement of the Cantor set. This should be essentially immediate by construction.
$endgroup$
– Andrés E. Caicedo
Jan 3 at 20:34
add a comment |
6
$begingroup$
Check that the Cantor function is constant on each connected component of the complement of the Cantor set. This should be essentially immediate by construction.
$endgroup$
– Andrés E. Caicedo
Jan 3 at 20:34
6
6
$begingroup$
Check that the Cantor function is constant on each connected component of the complement of the Cantor set. This should be essentially immediate by construction.
$endgroup$
– Andrés E. Caicedo
Jan 3 at 20:34
$begingroup$
Check that the Cantor function is constant on each connected component of the complement of the Cantor set. This should be essentially immediate by construction.
$endgroup$
– Andrés E. Caicedo
Jan 3 at 20:34
add a comment |
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$begingroup$
Check that the Cantor function is constant on each connected component of the complement of the Cantor set. This should be essentially immediate by construction.
$endgroup$
– Andrés E. Caicedo
Jan 3 at 20:34