Confusion regarding Cauchy's General Principle and Uniform Convergence
$begingroup$
The definitions of the two are so alike, that it confuses me.
Cauchy's General Principle:
The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x to a$, is that:
If $forall epsilon>0$, $exists$ a $ delta>0$ such that $|f(x_1)-f(x_2)| < epsilon$ whenever $0<|x_1-a|<delta, 0<|x_2-a|<delta $
Uniform Continuity:
A function $f$ defined on a domain $D$ is said to be uniformly continuous on the set $S$ $( S subset D)$ if:
$forall epsilon >0$, $exists $ a $delta>0$, such that $|f(x_1)-f(x_2)|< epsilon$ for any two points $x_1, x_2 in S$ with $|x_1-x_2| < delta$.
First of all, I understand that the former definition is a "Local Concept" i.e. it focuses around the particular point $a$ , while the other one is a "Global Concept", i.e. defined all over the set $S$. Another thing is, the sign "$0<$ " in the General Principle. It deletes the point "$a$" from the neighbourhood of both $x_1$ and $x_2$, while the other does not.
But the similarity is, both definitions give independence to choose arbitrary pair of points within a certain neighbourhood.
Is anything wrong with my perception? Am I missing anything very obvious?
Please do provide further insight.
limits continuity definition cauchy-sequences uniform-continuity
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
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add a comment |
$begingroup$
The definitions of the two are so alike, that it confuses me.
Cauchy's General Principle:
The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x to a$, is that:
If $forall epsilon>0$, $exists$ a $ delta>0$ such that $|f(x_1)-f(x_2)| < epsilon$ whenever $0<|x_1-a|<delta, 0<|x_2-a|<delta $
Uniform Continuity:
A function $f$ defined on a domain $D$ is said to be uniformly continuous on the set $S$ $( S subset D)$ if:
$forall epsilon >0$, $exists $ a $delta>0$, such that $|f(x_1)-f(x_2)|< epsilon$ for any two points $x_1, x_2 in S$ with $|x_1-x_2| < delta$.
First of all, I understand that the former definition is a "Local Concept" i.e. it focuses around the particular point $a$ , while the other one is a "Global Concept", i.e. defined all over the set $S$. Another thing is, the sign "$0<$ " in the General Principle. It deletes the point "$a$" from the neighbourhood of both $x_1$ and $x_2$, while the other does not.
But the similarity is, both definitions give independence to choose arbitrary pair of points within a certain neighbourhood.
Is anything wrong with my perception? Am I missing anything very obvious?
Please do provide further insight.
limits continuity definition cauchy-sequences uniform-continuity
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
$endgroup$
add a comment |
$begingroup$
The definitions of the two are so alike, that it confuses me.
Cauchy's General Principle:
The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x to a$, is that:
If $forall epsilon>0$, $exists$ a $ delta>0$ such that $|f(x_1)-f(x_2)| < epsilon$ whenever $0<|x_1-a|<delta, 0<|x_2-a|<delta $
Uniform Continuity:
A function $f$ defined on a domain $D$ is said to be uniformly continuous on the set $S$ $( S subset D)$ if:
$forall epsilon >0$, $exists $ a $delta>0$, such that $|f(x_1)-f(x_2)|< epsilon$ for any two points $x_1, x_2 in S$ with $|x_1-x_2| < delta$.
First of all, I understand that the former definition is a "Local Concept" i.e. it focuses around the particular point $a$ , while the other one is a "Global Concept", i.e. defined all over the set $S$. Another thing is, the sign "$0<$ " in the General Principle. It deletes the point "$a$" from the neighbourhood of both $x_1$ and $x_2$, while the other does not.
But the similarity is, both definitions give independence to choose arbitrary pair of points within a certain neighbourhood.
Is anything wrong with my perception? Am I missing anything very obvious?
Please do provide further insight.
limits continuity definition cauchy-sequences uniform-continuity
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
$endgroup$
The definitions of the two are so alike, that it confuses me.
Cauchy's General Principle:
The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x to a$, is that:
If $forall epsilon>0$, $exists$ a $ delta>0$ such that $|f(x_1)-f(x_2)| < epsilon$ whenever $0<|x_1-a|<delta, 0<|x_2-a|<delta $
Uniform Continuity:
A function $f$ defined on a domain $D$ is said to be uniformly continuous on the set $S$ $( S subset D)$ if:
$forall epsilon >0$, $exists $ a $delta>0$, such that $|f(x_1)-f(x_2)|< epsilon$ for any two points $x_1, x_2 in S$ with $|x_1-x_2| < delta$.
First of all, I understand that the former definition is a "Local Concept" i.e. it focuses around the particular point $a$ , while the other one is a "Global Concept", i.e. defined all over the set $S$. Another thing is, the sign "$0<$ " in the General Principle. It deletes the point "$a$" from the neighbourhood of both $x_1$ and $x_2$, while the other does not.
But the similarity is, both definitions give independence to choose arbitrary pair of points within a certain neighbourhood.
Is anything wrong with my perception? Am I missing anything very obvious?
Please do provide further insight.
limits continuity definition cauchy-sequences uniform-continuity
limits continuity definition cauchy-sequences uniform-continuity
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
asked Jan 2 at 10:20
Subhasis BiswasSubhasis Biswas
537512
537512
add a comment |
add a comment |
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I get it that way:
simple convergence :
given a $in$ I, $forall epsilon>0, exists delta$ such as $|x-a|<delta implies |f(x)-f(a)|<epsilon$
therefore this location is local, we defined a before epsilon so every epsilon is a function of a, and every delta a function of epsilon.
In your definition of uniform continuity, x1 and x2 are introduced after epsilon and delta, (not x dependents) which guarantees the fonction converges at every x at the same "speed".
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
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$begingroup$
I get it that way:
simple convergence :
given a $in$ I, $forall epsilon>0, exists delta$ such as $|x-a|<delta implies |f(x)-f(a)|<epsilon$
therefore this location is local, we defined a before epsilon so every epsilon is a function of a, and every delta a function of epsilon.
In your definition of uniform continuity, x1 and x2 are introduced after epsilon and delta, (not x dependents) which guarantees the fonction converges at every x at the same "speed".
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
$endgroup$
add a comment |
$begingroup$
I get it that way:
simple convergence :
given a $in$ I, $forall epsilon>0, exists delta$ such as $|x-a|<delta implies |f(x)-f(a)|<epsilon$
therefore this location is local, we defined a before epsilon so every epsilon is a function of a, and every delta a function of epsilon.
In your definition of uniform continuity, x1 and x2 are introduced after epsilon and delta, (not x dependents) which guarantees the fonction converges at every x at the same "speed".
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
$endgroup$
add a comment |
$begingroup$
I get it that way:
simple convergence :
given a $in$ I, $forall epsilon>0, exists delta$ such as $|x-a|<delta implies |f(x)-f(a)|<epsilon$
therefore this location is local, we defined a before epsilon so every epsilon is a function of a, and every delta a function of epsilon.
In your definition of uniform continuity, x1 and x2 are introduced after epsilon and delta, (not x dependents) which guarantees the fonction converges at every x at the same "speed".
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
$endgroup$
I get it that way:
simple convergence :
given a $in$ I, $forall epsilon>0, exists delta$ such as $|x-a|<delta implies |f(x)-f(a)|<epsilon$
therefore this location is local, we defined a before epsilon so every epsilon is a function of a, and every delta a function of epsilon.
In your definition of uniform continuity, x1 and x2 are introduced after epsilon and delta, (not x dependents) which guarantees the fonction converges at every x at the same "speed".
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
answered Jan 2 at 10:54
ReSpir3ReSpir3
1
1
add a comment |
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