Set of nowhere differentiable function in C[(0,1]) is dense
How can we prove with Baire's Theorem that in C[(0,1]), the set of nowhere differentiable function is dense.
normed-spaces
asked Nov 25 at 3:25
mimi
175
add a comment |
How can we prove with Baire's Theorem that in C[(0,1]), the set of nowhere differentiable function is dense.
normed-spaces
asked Nov 25 at 3:25
mimi
175
I have sketched the proof below but if you show some work, you will get a lot more feedback here. Otherwise, prepare to be flooded with downvotes.
– Matematleta
Nov 25 at 4:04
add a comment |
How can we prove with Baire's Theorem that in C[(0,1]), the set of nowhere differentiable function is dense.
normed-spaces
asked Nov 25 at 3:25
mimi
175
How can we prove with Baire's Theorem that in C[(0,1]), the set of nowhere differentiable function is dense.
normed-spaces
normed-spaces
asked Nov 25 at 3:25
mimi
175
asked Nov 25 at 3:25
mimi
175
asked Nov 25 at 3:25
mimi
175
asked Nov 25 at 3:25
mimi
175
asked Nov 25 at 3:25
mimi
175
175
I have sketched the proof below but if you show some work, you will get a lot more feedback here. Otherwise, prepare to be flooded with downvotes.
– Matematleta
Nov 25 at 4:04
add a comment |
I have sketched the proof below but if you show some work, you will get a lot more feedback here. Otherwise, prepare to be flooded with downvotes.
– Matematleta
Nov 25 at 4:04
I have sketched the proof below but if you show some work, you will get a lot more feedback here. Otherwise, prepare to be flooded with downvotes.
– Matematleta
Nov 25 at 4:04
I have sketched the proof below but if you show some work, you will get a lot more feedback here. Otherwise, prepare to be flooded with downvotes.
– Matematleta
Nov 25 at 4:04
add a comment |
1 Answer
1
active
oldest
votes
Here is a sketch:
$1).$ Show that $C([0,1])$ is not the countable union of nowhere dense sets.
$2).$ Set
$D={fin C([0,1]): f text{ is differentiable at some $x$ }}$
and
$A_{n,m}=left { fin C([0,1]): frac{f(t)-f(x)}{t-x})<n text{if} 0<|x-t|<frac{1}{m}right }$.
$3). Dsubseteq A_{n,m}$ and $A_{n,m}$ is closed.
$4). A_{n,m}$ are nowhere dense. (This is the only really hard part).
$5).$ Conclude.
answered Nov 25 at 4:01
Matematleta
9,9522918
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Here is a sketch:
$1).$ Show that $C([0,1])$ is not the countable union of nowhere dense sets.
$2).$ Set
$D={fin C([0,1]): f text{ is differentiable at some $x$ }}$
and
$A_{n,m}=left { fin C([0,1]): frac{f(t)-f(x)}{t-x})<n text{if} 0<|x-t|<frac{1}{m}right }$.
$3). Dsubseteq A_{n,m}$ and $A_{n,m}$ is closed.
$4). A_{n,m}$ are nowhere dense. (This is the only really hard part).
$5).$ Conclude.
answered Nov 25 at 4:01
Matematleta
9,9522918
add a comment |
Here is a sketch:
$1).$ Show that $C([0,1])$ is not the countable union of nowhere dense sets.
$2).$ Set
$D={fin C([0,1]): f text{ is differentiable at some $x$ }}$
and
$A_{n,m}=left { fin C([0,1]): frac{f(t)-f(x)}{t-x})<n text{if} 0<|x-t|<frac{1}{m}right }$.
$3). Dsubseteq A_{n,m}$ and $A_{n,m}$ is closed.
$4). A_{n,m}$ are nowhere dense. (This is the only really hard part).
$5).$ Conclude.
answered Nov 25 at 4:01
Matematleta
9,9522918
add a comment |
Here is a sketch:
$1).$ Show that $C([0,1])$ is not the countable union of nowhere dense sets.
$2).$ Set
$D={fin C([0,1]): f text{ is differentiable at some $x$ }}$
and
$A_{n,m}=left { fin C([0,1]): frac{f(t)-f(x)}{t-x})<n text{if} 0<|x-t|<frac{1}{m}right }$.
$3). Dsubseteq A_{n,m}$ and $A_{n,m}$ is closed.
$4). A_{n,m}$ are nowhere dense. (This is the only really hard part).
$5).$ Conclude.
answered Nov 25 at 4:01
Matematleta
9,9522918
Here is a sketch:
$1).$ Show that $C([0,1])$ is not the countable union of nowhere dense sets.
$2).$ Set
$D={fin C([0,1]): f text{ is differentiable at some $x$ }}$
and
$A_{n,m}=left { fin C([0,1]): frac{f(t)-f(x)}{t-x})<n text{if} 0<|x-t|<frac{1}{m}right }$.
$3). Dsubseteq A_{n,m}$ and $A_{n,m}$ is closed.
$4). A_{n,m}$ are nowhere dense. (This is the only really hard part).
$5).$ Conclude.
answered Nov 25 at 4:01
Matematleta
9,9522918
answered Nov 25 at 4:01
Matematleta
9,9522918
answered Nov 25 at 4:01
Matematleta
9,9522918
answered Nov 25 at 4:01
Matematleta
9,9522918
9,9522918
add a comment |
add a comment |
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I have sketched the proof below but if you show some work, you will get a lot more feedback here. Otherwise, prepare to be flooded with downvotes.
– Matematleta
Nov 25 at 4:04