Show that $|f(x)-L|leq K|x-c| implieslim_{xto c} f(x)=L$
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Let $I$ be an interval in $mathbb{R}$ such that $f:Irightarrowmathbb{R}$ is a function and let $cin I.$
Show that $|f(x)-L|leq K|x-c| implieslim_{xto c} f(x)=L$
Proof:
Suppose w.l.o.g. that $K>0$. Then we have $K|x-c|>0$ since $|x-c|>0$. Now since $xin I$ and $cin I$, $existsepsilon>0$ S.t. $|x-c|<frac{epsilon}{K}$; we can make the distance between $x$ and $c$ as small as we like. Therefore by assumption, $$|f(x)-L|<K|x-c|<K(frac{epsilon}{K})=epsilon$$
Whence, we conclude since $epsilon$ was arbitrary, that $lim_{xto c}f(x)=L$. $blacksquare$
Is this proof valid?
real-analysis limits proof-verification lipschitz-functions
add a comment |
up vote
1
down vote
favorite
Let $I$ be an interval in $mathbb{R}$ such that $f:Irightarrowmathbb{R}$ is a function and let $cin I.$
Show that $|f(x)-L|leq K|x-c| implieslim_{xto c} f(x)=L$
Proof:
Suppose w.l.o.g. that $K>0$. Then we have $K|x-c|>0$ since $|x-c|>0$. Now since $xin I$ and $cin I$, $existsepsilon>0$ S.t. $|x-c|<frac{epsilon}{K}$; we can make the distance between $x$ and $c$ as small as we like. Therefore by assumption, $$|f(x)-L|<K|x-c|<K(frac{epsilon}{K})=epsilon$$
Whence, we conclude since $epsilon$ was arbitrary, that $lim_{xto c}f(x)=L$. $blacksquare$
Is this proof valid?
real-analysis limits proof-verification lipschitz-functions
Realize that $K$ cannot be negative. This is a Lipschitz-type inequality.
– Sean Roberson
Nov 17 at 0:10
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $I$ be an interval in $mathbb{R}$ such that $f:Irightarrowmathbb{R}$ is a function and let $cin I.$
Show that $|f(x)-L|leq K|x-c| implieslim_{xto c} f(x)=L$
Proof:
Suppose w.l.o.g. that $K>0$. Then we have $K|x-c|>0$ since $|x-c|>0$. Now since $xin I$ and $cin I$, $existsepsilon>0$ S.t. $|x-c|<frac{epsilon}{K}$; we can make the distance between $x$ and $c$ as small as we like. Therefore by assumption, $$|f(x)-L|<K|x-c|<K(frac{epsilon}{K})=epsilon$$
Whence, we conclude since $epsilon$ was arbitrary, that $lim_{xto c}f(x)=L$. $blacksquare$
Is this proof valid?
real-analysis limits proof-verification lipschitz-functions
Let $I$ be an interval in $mathbb{R}$ such that $f:Irightarrowmathbb{R}$ is a function and let $cin I.$
Show that $|f(x)-L|leq K|x-c| implieslim_{xto c} f(x)=L$
Proof:
Suppose w.l.o.g. that $K>0$. Then we have $K|x-c|>0$ since $|x-c|>0$. Now since $xin I$ and $cin I$, $existsepsilon>0$ S.t. $|x-c|<frac{epsilon}{K}$; we can make the distance between $x$ and $c$ as small as we like. Therefore by assumption, $$|f(x)-L|<K|x-c|<K(frac{epsilon}{K})=epsilon$$
Whence, we conclude since $epsilon$ was arbitrary, that $lim_{xto c}f(x)=L$. $blacksquare$
Is this proof valid?
real-analysis limits proof-verification lipschitz-functions
real-analysis limits proof-verification lipschitz-functions
edited Nov 17 at 0:28
Scientifica
6,04141332
6,04141332
asked Nov 17 at 0:04
coreyman317
681218
681218
Realize that $K$ cannot be negative. This is a Lipschitz-type inequality.
– Sean Roberson
Nov 17 at 0:10
add a comment |
Realize that $K$ cannot be negative. This is a Lipschitz-type inequality.
– Sean Roberson
Nov 17 at 0:10
Realize that $K$ cannot be negative. This is a Lipschitz-type inequality.
– Sean Roberson
Nov 17 at 0:10
Realize that $K$ cannot be negative. This is a Lipschitz-type inequality.
– Sean Roberson
Nov 17 at 0:10
add a comment |
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
You have the right ideas, well done! But unfortunately there are some points in the proof that are wrong.
You say that $K|x-c|>0$ because $|x-c|>0$. But if you take $x=c$, this is no longer true.
You also say that $existsvarepsilon>0,|x-c|<dfrac{varepsilon}{K}$. This statement is true, but it doesn't mean that $varepsilon$ can be as small as you want unlike what you said. In fact, such an $varepsilon$ has to satisfy $K|x-c|<varepsilon$, so epsilon can't be as small as, say, $frac{1}{2}K|x-c|$. The statement also is of the form $existsvarepsilon$, so $varepsilon$ is not arbitrary, but rather specifically chosen.
Here's what you want to prove:
$$forallvarepsilon>0,existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
So the natural way to start your proof is: Let $varepsilon>0$. Now that's a truly arbitrary. So what you need to prove is this:
$$existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
And here, your choice of $delta=dfrac{varepsilon}{K}$ does the work.
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
1
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
add a comment |
up vote
1
down vote
Yes it is valid once you take into account Sean Roberson's comment that $K>0$. Then for $f:I rightarrow mathbb{R}$ such that for some $c in I$ and some $L in mathbb{R}$, we have that $|f(x) - L| leq K |x - c|$. For any $epsilon > 0$, choose any $x in I$ close enough to $c$ such that $|x - c| < epsilon$. Then
$$|f(x) - L| leq K |x - c| < Kcdot epsilon.
$$
Now let us set $tilde{epsilon} := Kcdot epsilon > 0$, then for any $x$ close enough to $c$ such that $| x - c| < epsilon$, this implies that $|f(x) - L| < tilde{epsilon}$.
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
You have the right ideas, well done! But unfortunately there are some points in the proof that are wrong.
You say that $K|x-c|>0$ because $|x-c|>0$. But if you take $x=c$, this is no longer true.
You also say that $existsvarepsilon>0,|x-c|<dfrac{varepsilon}{K}$. This statement is true, but it doesn't mean that $varepsilon$ can be as small as you want unlike what you said. In fact, such an $varepsilon$ has to satisfy $K|x-c|<varepsilon$, so epsilon can't be as small as, say, $frac{1}{2}K|x-c|$. The statement also is of the form $existsvarepsilon$, so $varepsilon$ is not arbitrary, but rather specifically chosen.
Here's what you want to prove:
$$forallvarepsilon>0,existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
So the natural way to start your proof is: Let $varepsilon>0$. Now that's a truly arbitrary. So what you need to prove is this:
$$existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
And here, your choice of $delta=dfrac{varepsilon}{K}$ does the work.
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
1
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
add a comment |
up vote
2
down vote
accepted
You have the right ideas, well done! But unfortunately there are some points in the proof that are wrong.
You say that $K|x-c|>0$ because $|x-c|>0$. But if you take $x=c$, this is no longer true.
You also say that $existsvarepsilon>0,|x-c|<dfrac{varepsilon}{K}$. This statement is true, but it doesn't mean that $varepsilon$ can be as small as you want unlike what you said. In fact, such an $varepsilon$ has to satisfy $K|x-c|<varepsilon$, so epsilon can't be as small as, say, $frac{1}{2}K|x-c|$. The statement also is of the form $existsvarepsilon$, so $varepsilon$ is not arbitrary, but rather specifically chosen.
Here's what you want to prove:
$$forallvarepsilon>0,existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
So the natural way to start your proof is: Let $varepsilon>0$. Now that's a truly arbitrary. So what you need to prove is this:
$$existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
And here, your choice of $delta=dfrac{varepsilon}{K}$ does the work.
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
1
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
You have the right ideas, well done! But unfortunately there are some points in the proof that are wrong.
You say that $K|x-c|>0$ because $|x-c|>0$. But if you take $x=c$, this is no longer true.
You also say that $existsvarepsilon>0,|x-c|<dfrac{varepsilon}{K}$. This statement is true, but it doesn't mean that $varepsilon$ can be as small as you want unlike what you said. In fact, such an $varepsilon$ has to satisfy $K|x-c|<varepsilon$, so epsilon can't be as small as, say, $frac{1}{2}K|x-c|$. The statement also is of the form $existsvarepsilon$, so $varepsilon$ is not arbitrary, but rather specifically chosen.
Here's what you want to prove:
$$forallvarepsilon>0,existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
So the natural way to start your proof is: Let $varepsilon>0$. Now that's a truly arbitrary. So what you need to prove is this:
$$existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
And here, your choice of $delta=dfrac{varepsilon}{K}$ does the work.
You have the right ideas, well done! But unfortunately there are some points in the proof that are wrong.
You say that $K|x-c|>0$ because $|x-c|>0$. But if you take $x=c$, this is no longer true.
You also say that $existsvarepsilon>0,|x-c|<dfrac{varepsilon}{K}$. This statement is true, but it doesn't mean that $varepsilon$ can be as small as you want unlike what you said. In fact, such an $varepsilon$ has to satisfy $K|x-c|<varepsilon$, so epsilon can't be as small as, say, $frac{1}{2}K|x-c|$. The statement also is of the form $existsvarepsilon$, so $varepsilon$ is not arbitrary, but rather specifically chosen.
Here's what you want to prove:
$$forallvarepsilon>0,existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
So the natural way to start your proof is: Let $varepsilon>0$. Now that's a truly arbitrary. So what you need to prove is this:
$$existsdelta>0,forall xin I,|x-c|<deltaimplies |f(x)-L|<varepsilon.$$
And here, your choice of $delta=dfrac{varepsilon}{K}$ does the work.
answered Nov 17 at 0:28
Scientifica
6,04141332
6,04141332
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
1
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
add a comment |
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
1
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
I thought in the definition of a limit, we take $c$ to be a cluster point which implies $xneq c$?
– coreyman317
Nov 17 at 0:30
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
@coreyman317 You're right, but what you wrote is $cin I$ and $xin I$ so $|x-c|>0$, which is wrong because we can take $x=c$ without any problem.
– Scientifica
Nov 17 at 0:37
1
1
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
@coreyman317 Let me correct my comment. You're right in saying that we take $c$ to be a cluster point, which implies that no matter how close you get to $c$, you can take $xneq c$. But you can also pick $x=c$. On the other hand, if you explicitly write "now let $xin Ibackslash{c}$", then indeed $xneq c$. By the way, I just saw that some define limit as: $$forallvarepsilon>0,existsdelta>0,forall xin I,0<|x-c|<deltaimplies |f(x)-L|<varepsilon,$$ which makes more sense tbh. If that's the case, then there's no need to assume that $|x-c|>0$.
– Scientifica
Nov 17 at 0:46
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
Thank you @Scientifica
– coreyman317
Nov 17 at 1:01
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
@coreyman317 It's a pleasure :)
– Scientifica
Nov 17 at 11:59
add a comment |
up vote
1
down vote
Yes it is valid once you take into account Sean Roberson's comment that $K>0$. Then for $f:I rightarrow mathbb{R}$ such that for some $c in I$ and some $L in mathbb{R}$, we have that $|f(x) - L| leq K |x - c|$. For any $epsilon > 0$, choose any $x in I$ close enough to $c$ such that $|x - c| < epsilon$. Then
$$|f(x) - L| leq K |x - c| < Kcdot epsilon.
$$
Now let us set $tilde{epsilon} := Kcdot epsilon > 0$, then for any $x$ close enough to $c$ such that $| x - c| < epsilon$, this implies that $|f(x) - L| < tilde{epsilon}$.
add a comment |
up vote
1
down vote
Yes it is valid once you take into account Sean Roberson's comment that $K>0$. Then for $f:I rightarrow mathbb{R}$ such that for some $c in I$ and some $L in mathbb{R}$, we have that $|f(x) - L| leq K |x - c|$. For any $epsilon > 0$, choose any $x in I$ close enough to $c$ such that $|x - c| < epsilon$. Then
$$|f(x) - L| leq K |x - c| < Kcdot epsilon.
$$
Now let us set $tilde{epsilon} := Kcdot epsilon > 0$, then for any $x$ close enough to $c$ such that $| x - c| < epsilon$, this implies that $|f(x) - L| < tilde{epsilon}$.
add a comment |
up vote
1
down vote
up vote
1
down vote
Yes it is valid once you take into account Sean Roberson's comment that $K>0$. Then for $f:I rightarrow mathbb{R}$ such that for some $c in I$ and some $L in mathbb{R}$, we have that $|f(x) - L| leq K |x - c|$. For any $epsilon > 0$, choose any $x in I$ close enough to $c$ such that $|x - c| < epsilon$. Then
$$|f(x) - L| leq K |x - c| < Kcdot epsilon.
$$
Now let us set $tilde{epsilon} := Kcdot epsilon > 0$, then for any $x$ close enough to $c$ such that $| x - c| < epsilon$, this implies that $|f(x) - L| < tilde{epsilon}$.
Yes it is valid once you take into account Sean Roberson's comment that $K>0$. Then for $f:I rightarrow mathbb{R}$ such that for some $c in I$ and some $L in mathbb{R}$, we have that $|f(x) - L| leq K |x - c|$. For any $epsilon > 0$, choose any $x in I$ close enough to $c$ such that $|x - c| < epsilon$. Then
$$|f(x) - L| leq K |x - c| < Kcdot epsilon.
$$
Now let us set $tilde{epsilon} := Kcdot epsilon > 0$, then for any $x$ close enough to $c$ such that $| x - c| < epsilon$, this implies that $|f(x) - L| < tilde{epsilon}$.
answered Nov 17 at 0:30
BenCWBrown
3557
3557
add a comment |
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Realize that $K$ cannot be negative. This is a Lipschitz-type inequality.
– Sean Roberson
Nov 17 at 0:10