Mixing
Existence of a function continuous everywhere and nowhere differentialable
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We know there exists some functions, such that Weierstrass one, that are continuous everywhere on $[0 ; 1]$ and nowhere differentiable. Their expression (and a little bit of work) would yield a proof.
However, I was wondering if we can just prove the existence of such a function, without finding it.
I searched on the web and found nothing. Would some of you have a reference or a proof?
Thanks in advance!
continuity differential-topology
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
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add a comment |
$begingroup$
We know there exists some functions, such that Weierstrass one, that are continuous everywhere on $[0 ; 1]$ and nowhere differentiable. Their expression (and a little bit of work) would yield a proof.
However, I was wondering if we can just prove the existence of such a function, without finding it.
I searched on the web and found nothing. Would some of you have a reference or a proof?
Thanks in advance!
continuity differential-topology
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
$endgroup$
$begingroup$
It's been a while, but I think this is a case where Baire category theorems come into play.
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– Cameron Williams
Jan 16 at 21:11
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Thank you I'll check that.
$endgroup$
– Cauchy is my master
Jan 16 at 21:17
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There is an example using the Baire Category Theorem in "A Primer of Real Functions" by Ralph Boas.
$endgroup$
– awkward
Jan 17 at 15:32
add a comment |
$begingroup$
We know there exists some functions, such that Weierstrass one, that are continuous everywhere on $[0 ; 1]$ and nowhere differentiable. Their expression (and a little bit of work) would yield a proof.
However, I was wondering if we can just prove the existence of such a function, without finding it.
I searched on the web and found nothing. Would some of you have a reference or a proof?
Thanks in advance!
continuity differential-topology
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
$endgroup$
We know there exists some functions, such that Weierstrass one, that are continuous everywhere on $[0 ; 1]$ and nowhere differentiable. Their expression (and a little bit of work) would yield a proof.
However, I was wondering if we can just prove the existence of such a function, without finding it.
I searched on the web and found nothing. Would some of you have a reference or a proof?
Thanks in advance!
continuity differential-topology
continuity differential-topology
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
asked Jan 16 at 21:09
Cauchy is my masterCauchy is my master
413
413
$begingroup$
It's been a while, but I think this is a case where Baire category theorems come into play.
$endgroup$
– Cameron Williams
Jan 16 at 21:11
$begingroup$
Thank you I'll check that.
$endgroup$
– Cauchy is my master
Jan 16 at 21:17
$begingroup$
There is an example using the Baire Category Theorem in "A Primer of Real Functions" by Ralph Boas.
$endgroup$
– awkward
Jan 17 at 15:32
add a comment |
$begingroup$
It's been a while, but I think this is a case where Baire category theorems come into play.
$endgroup$
– Cameron Williams
Jan 16 at 21:11
$begingroup$
Thank you I'll check that.
$endgroup$
– Cauchy is my master
Jan 16 at 21:17
$begingroup$
There is an example using the Baire Category Theorem in "A Primer of Real Functions" by Ralph Boas.
$endgroup$
– awkward
Jan 17 at 15:32
$begingroup$
It's been a while, but I think this is a case where Baire category theorems come into play.
$endgroup$
– Cameron Williams
Jan 16 at 21:11
$begingroup$
It's been a while, but I think this is a case where Baire category theorems come into play.
$endgroup$
– Cameron Williams
Jan 16 at 21:11
$begingroup$
Thank you I'll check that.
$endgroup$
– Cauchy is my master
Jan 16 at 21:17
$begingroup$
Thank you I'll check that.
$endgroup$
– Cauchy is my master
Jan 16 at 21:17
$begingroup$
There is an example using the Baire Category Theorem in "A Primer of Real Functions" by Ralph Boas.
$endgroup$
– awkward
Jan 17 at 15:32
$begingroup$
There is an example using the Baire Category Theorem in "A Primer of Real Functions" by Ralph Boas.
$endgroup$
– awkward
Jan 17 at 15:32
add a comment |
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$begingroup$
It's been a while, but I think this is a case where Baire category theorems come into play.
$endgroup$
– Cameron Williams
Jan 16 at 21:11
$begingroup$
Thank you I'll check that.
$endgroup$
– Cauchy is my master
Jan 16 at 21:17
$begingroup$
There is an example using the Baire Category Theorem in "A Primer of Real Functions" by Ralph Boas.
$endgroup$
– awkward
Jan 17 at 15:32