Need Help : Proving polynomials are continuous, without circular reasoning
Clash Royale CLAN TAG#URR8PPP
$begingroup$
I know there are a lot of answers regarding continuity of polynomials. But, this question is different.
We need to have $ lim_xto a x^n = a^n$ , $n in N$ , to be able to prove that polynomials are continuous. This fact is derived from the product rule (or may be it can't be, which is my question). The product rule is proved using square roots, so it assumes the existence of square roots. The fact that For every non negative number $x$, it's $n^th$ root exists, i.e. $x^n$ is invertible, assumes the continuity of $x^n$ because this is proven using Intermediate value Theorem.
Bam - Circular reasoning ! Or am I Wrong ?
Here's the only proof of product rule which I know :
Assume $ lim_xto a f(x) = L$ and $ lim_xto a g(x) = K$
Let $ϵ > 0$ be any positive number
Hence, $∃delta_1> 0 ∶ 0<|x-a|<δ_1⟹|f(x)-L|<sqrtepsilon$
And $∃δ_2>0 ∶ 0<|x-a|<δ_2 ⟹ |g(x)-K|<sqrtepsilon$
Let $δ=minδ_1,δ_2$
Hence, $0<|x-a|<δ$
⟹$|(f(x)-L)(g(x)-K)-0|<sqrtepsilon sqrtepsilon = ϵ$
Hence, $lim_x to a (f(x)-L)(g(x)-K) = 0$
⟹$lim_x to a (f(x)g(x)-Kf(x)-Lg(x)+KL) = 0$
And then the result follows.
Even if we use $epsilon$ in place of $√ϵ$ , we end up with
$0<|x-a|<δ⟹|(f(x)-L)(g(x)-K)-0|<epsiloncdot epsilon = ϵ^2$ , and then we have to prove that the range of $epsilon^2$ is $[0,infty]$, which amounts to proving that for each number in $[0,infty]$ , a corresponding square root exists.
real-analysis limits continuity epsilon-delta
$endgroup$
|
show 6 more comments
$begingroup$
I know there are a lot of answers regarding continuity of polynomials. But, this question is different.
We need to have $ lim_xto a x^n = a^n$ , $n in N$ , to be able to prove that polynomials are continuous. This fact is derived from the product rule (or may be it can't be, which is my question). The product rule is proved using square roots, so it assumes the existence of square roots. The fact that For every non negative number $x$, it's $n^th$ root exists, i.e. $x^n$ is invertible, assumes the continuity of $x^n$ because this is proven using Intermediate value Theorem.
Bam - Circular reasoning ! Or am I Wrong ?
Here's the only proof of product rule which I know :
Assume $ lim_xto a f(x) = L$ and $ lim_xto a g(x) = K$
Let $ϵ > 0$ be any positive number
Hence, $∃delta_1> 0 ∶ 0<|x-a|<δ_1⟹|f(x)-L|<sqrtepsilon$
And $∃δ_2>0 ∶ 0<|x-a|<δ_2 ⟹ |g(x)-K|<sqrtepsilon$
Let $δ=minδ_1,δ_2$
Hence, $0<|x-a|<δ$
⟹$|(f(x)-L)(g(x)-K)-0|<sqrtepsilon sqrtepsilon = ϵ$
Hence, $lim_x to a (f(x)-L)(g(x)-K) = 0$
⟹$lim_x to a (f(x)g(x)-Kf(x)-Lg(x)+KL) = 0$
And then the result follows.
Even if we use $epsilon$ in place of $√ϵ$ , we end up with
$0<|x-a|<δ⟹|(f(x)-L)(g(x)-K)-0|<epsiloncdot epsilon = ϵ^2$ , and then we have to prove that the range of $epsilon^2$ is $[0,infty]$, which amounts to proving that for each number in $[0,infty]$ , a corresponding square root exists.
real-analysis limits continuity epsilon-delta
$endgroup$
1
$begingroup$
You can use MathJax in a single formula - you don't have to write$f(x)$<$g(x)$
and similar formulas. Also I see two different notations $lim_xto a$ and $Lim_xto a$ in the post - but without an explanation whether they are intended to represent two different things or they are both just different notations for the usual limit.
$endgroup$
– Martin Sleziak
Jan 7 at 8:57
$begingroup$
Thank you. I have edited the question. The limit with lowercase L and uppercase L are not different, it was my mistake.
$endgroup$
– Steve
Jan 7 at 9:01
2
$begingroup$
Why don't you use the fact that $$|f(x) g(x) - LK|leq |g(x) ||f(x) - L|+|L||g(x) - K|$$ and show that each term on right can be made less than $epsilon /2$?
$endgroup$
– Paramanand Singh
Jan 7 at 9:10
1
$begingroup$
Use the fact that $g(x) to K$ and hence $|g(x) |<|K|+1$ and we can take $|f(x) - L|<epsilon /(2(|K|+1))$. This handles first term. In similar manner you can take $|g(x) - K|<epsilon/(2(|L|+1))$ to handle second term.
$endgroup$
– Paramanand Singh
Jan 7 at 9:29
2
$begingroup$
In general when you are dealing with $epsilon, delta$ proofs the expression based on these $epsilon, delta$ should not involve operations other than $+, -, times, /$.
$endgroup$
– Paramanand Singh
Jan 7 at 9:32
|
show 6 more comments
$begingroup$
I know there are a lot of answers regarding continuity of polynomials. But, this question is different.
We need to have $ lim_xto a x^n = a^n$ , $n in N$ , to be able to prove that polynomials are continuous. This fact is derived from the product rule (or may be it can't be, which is my question). The product rule is proved using square roots, so it assumes the existence of square roots. The fact that For every non negative number $x$, it's $n^th$ root exists, i.e. $x^n$ is invertible, assumes the continuity of $x^n$ because this is proven using Intermediate value Theorem.
Bam - Circular reasoning ! Or am I Wrong ?
Here's the only proof of product rule which I know :
Assume $ lim_xto a f(x) = L$ and $ lim_xto a g(x) = K$
Let $ϵ > 0$ be any positive number
Hence, $∃delta_1> 0 ∶ 0<|x-a|<δ_1⟹|f(x)-L|<sqrtepsilon$
And $∃δ_2>0 ∶ 0<|x-a|<δ_2 ⟹ |g(x)-K|<sqrtepsilon$
Let $δ=minδ_1,δ_2$
Hence, $0<|x-a|<δ$
⟹$|(f(x)-L)(g(x)-K)-0|<sqrtepsilon sqrtepsilon = ϵ$
Hence, $lim_x to a (f(x)-L)(g(x)-K) = 0$
⟹$lim_x to a (f(x)g(x)-Kf(x)-Lg(x)+KL) = 0$
And then the result follows.
Even if we use $epsilon$ in place of $√ϵ$ , we end up with
$0<|x-a|<δ⟹|(f(x)-L)(g(x)-K)-0|<epsiloncdot epsilon = ϵ^2$ , and then we have to prove that the range of $epsilon^2$ is $[0,infty]$, which amounts to proving that for each number in $[0,infty]$ , a corresponding square root exists.
real-analysis limits continuity epsilon-delta
$endgroup$
I know there are a lot of answers regarding continuity of polynomials. But, this question is different.
We need to have $ lim_xto a x^n = a^n$ , $n in N$ , to be able to prove that polynomials are continuous. This fact is derived from the product rule (or may be it can't be, which is my question). The product rule is proved using square roots, so it assumes the existence of square roots. The fact that For every non negative number $x$, it's $n^th$ root exists, i.e. $x^n$ is invertible, assumes the continuity of $x^n$ because this is proven using Intermediate value Theorem.
Bam - Circular reasoning ! Or am I Wrong ?
Here's the only proof of product rule which I know :
Assume $ lim_xto a f(x) = L$ and $ lim_xto a g(x) = K$
Let $ϵ > 0$ be any positive number
Hence, $∃delta_1> 0 ∶ 0<|x-a|<δ_1⟹|f(x)-L|<sqrtepsilon$
And $∃δ_2>0 ∶ 0<|x-a|<δ_2 ⟹ |g(x)-K|<sqrtepsilon$
Let $δ=minδ_1,δ_2$
Hence, $0<|x-a|<δ$
⟹$|(f(x)-L)(g(x)-K)-0|<sqrtepsilon sqrtepsilon = ϵ$
Hence, $lim_x to a (f(x)-L)(g(x)-K) = 0$
⟹$lim_x to a (f(x)g(x)-Kf(x)-Lg(x)+KL) = 0$
And then the result follows.
Even if we use $epsilon$ in place of $√ϵ$ , we end up with
$0<|x-a|<δ⟹|(f(x)-L)(g(x)-K)-0|<epsiloncdot epsilon = ϵ^2$ , and then we have to prove that the range of $epsilon^2$ is $[0,infty]$, which amounts to proving that for each number in $[0,infty]$ , a corresponding square root exists.
real-analysis limits continuity epsilon-delta
real-analysis limits continuity epsilon-delta
edited Jan 7 at 9:09
Martin Sleziak
44.7k9117272
44.7k9117272
asked Jan 7 at 8:39
SteveSteve
828
828
1
$begingroup$
You can use MathJax in a single formula - you don't have to write$f(x)$<$g(x)$
and similar formulas. Also I see two different notations $lim_xto a$ and $Lim_xto a$ in the post - but without an explanation whether they are intended to represent two different things or they are both just different notations for the usual limit.
$endgroup$
– Martin Sleziak
Jan 7 at 8:57
$begingroup$
Thank you. I have edited the question. The limit with lowercase L and uppercase L are not different, it was my mistake.
$endgroup$
– Steve
Jan 7 at 9:01
2
$begingroup$
Why don't you use the fact that $$|f(x) g(x) - LK|leq |g(x) ||f(x) - L|+|L||g(x) - K|$$ and show that each term on right can be made less than $epsilon /2$?
$endgroup$
– Paramanand Singh
Jan 7 at 9:10
1
$begingroup$
Use the fact that $g(x) to K$ and hence $|g(x) |<|K|+1$ and we can take $|f(x) - L|<epsilon /(2(|K|+1))$. This handles first term. In similar manner you can take $|g(x) - K|<epsilon/(2(|L|+1))$ to handle second term.
$endgroup$
– Paramanand Singh
Jan 7 at 9:29
2
$begingroup$
In general when you are dealing with $epsilon, delta$ proofs the expression based on these $epsilon, delta$ should not involve operations other than $+, -, times, /$.
$endgroup$
– Paramanand Singh
Jan 7 at 9:32
|
show 6 more comments
1
$begingroup$
You can use MathJax in a single formula - you don't have to write$f(x)$<$g(x)$
and similar formulas. Also I see two different notations $lim_xto a$ and $Lim_xto a$ in the post - but without an explanation whether they are intended to represent two different things or they are both just different notations for the usual limit.
$endgroup$
– Martin Sleziak
Jan 7 at 8:57
$begingroup$
Thank you. I have edited the question. The limit with lowercase L and uppercase L are not different, it was my mistake.
$endgroup$
– Steve
Jan 7 at 9:01
2
$begingroup$
Why don't you use the fact that $$|f(x) g(x) - LK|leq |g(x) ||f(x) - L|+|L||g(x) - K|$$ and show that each term on right can be made less than $epsilon /2$?
$endgroup$
– Paramanand Singh
Jan 7 at 9:10
1
$begingroup$
Use the fact that $g(x) to K$ and hence $|g(x) |<|K|+1$ and we can take $|f(x) - L|<epsilon /(2(|K|+1))$. This handles first term. In similar manner you can take $|g(x) - K|<epsilon/(2(|L|+1))$ to handle second term.
$endgroup$
– Paramanand Singh
Jan 7 at 9:29
2
$begingroup$
In general when you are dealing with $epsilon, delta$ proofs the expression based on these $epsilon, delta$ should not involve operations other than $+, -, times, /$.
$endgroup$
– Paramanand Singh
Jan 7 at 9:32
1
1
$begingroup$
You can use MathJax in a single formula - you don't have to write
$f(x)$<$g(x)$
and similar formulas. Also I see two different notations $lim_xto a$ and $Lim_xto a$ in the post - but without an explanation whether they are intended to represent two different things or they are both just different notations for the usual limit.$endgroup$
– Martin Sleziak
Jan 7 at 8:57
$begingroup$
You can use MathJax in a single formula - you don't have to write
$f(x)$<$g(x)$
and similar formulas. Also I see two different notations $lim_xto a$ and $Lim_xto a$ in the post - but without an explanation whether they are intended to represent two different things or they are both just different notations for the usual limit.$endgroup$
– Martin Sleziak
Jan 7 at 8:57
$begingroup$
Thank you. I have edited the question. The limit with lowercase L and uppercase L are not different, it was my mistake.
$endgroup$
– Steve
Jan 7 at 9:01
$begingroup$
Thank you. I have edited the question. The limit with lowercase L and uppercase L are not different, it was my mistake.
$endgroup$
– Steve
Jan 7 at 9:01
2
2
$begingroup$
Why don't you use the fact that $$|f(x) g(x) - LK|leq |g(x) ||f(x) - L|+|L||g(x) - K|$$ and show that each term on right can be made less than $epsilon /2$?
$endgroup$
– Paramanand Singh
Jan 7 at 9:10
$begingroup$
Why don't you use the fact that $$|f(x) g(x) - LK|leq |g(x) ||f(x) - L|+|L||g(x) - K|$$ and show that each term on right can be made less than $epsilon /2$?
$endgroup$
– Paramanand Singh
Jan 7 at 9:10
1
1
$begingroup$
Use the fact that $g(x) to K$ and hence $|g(x) |<|K|+1$ and we can take $|f(x) - L|<epsilon /(2(|K|+1))$. This handles first term. In similar manner you can take $|g(x) - K|<epsilon/(2(|L|+1))$ to handle second term.
$endgroup$
– Paramanand Singh
Jan 7 at 9:29
$begingroup$
Use the fact that $g(x) to K$ and hence $|g(x) |<|K|+1$ and we can take $|f(x) - L|<epsilon /(2(|K|+1))$. This handles first term. In similar manner you can take $|g(x) - K|<epsilon/(2(|L|+1))$ to handle second term.
$endgroup$
– Paramanand Singh
Jan 7 at 9:29
2
2
$begingroup$
In general when you are dealing with $epsilon, delta$ proofs the expression based on these $epsilon, delta$ should not involve operations other than $+, -, times, /$.
$endgroup$
– Paramanand Singh
Jan 7 at 9:32
$begingroup$
In general when you are dealing with $epsilon, delta$ proofs the expression based on these $epsilon, delta$ should not involve operations other than $+, -, times, /$.
$endgroup$
– Paramanand Singh
Jan 7 at 9:32
|
show 6 more comments
2 Answers
2
active
oldest
votes
$begingroup$
The proof that the product of continuous functions is also continuous does not need to use square roots. You could add the condition that $0 < epsilon < 1$, in which case $epsilon^2 < epsilon$. Or you could use a convergent sequence $x_n$ such that $displaystyle lim_n mathop to infty x_n = a$, and show that
$$lim_n to inftyf(x_n)g(x_n) = fleft( lim_n to infty x_n right)gleft(lim_n to infty x_nright) = f(a)g(a)$$
$endgroup$
add a comment |
$begingroup$
You don't need existence of square root. You just need to have that $forall epsilon > 0 exists epsilon^prime > 0$ s. t. $epsilon^prime^2 < epsilon$. Then, you'll have $|(f(x) - K)(g(x)-L)| < epsilon^prime^2 < epsilon$ which gives you what you need anyway. You don't need the bound $epsilon$ to actually be reachable.
For $epsilon < 1$, $epsilon$ itself can serve as $epsilon^prime$.
$endgroup$
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
add a comment |
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2 Answers
2
active
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votes
2 Answers
2
active
oldest
votes
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oldest
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active
oldest
votes
$begingroup$
The proof that the product of continuous functions is also continuous does not need to use square roots. You could add the condition that $0 < epsilon < 1$, in which case $epsilon^2 < epsilon$. Or you could use a convergent sequence $x_n$ such that $displaystyle lim_n mathop to infty x_n = a$, and show that
$$lim_n to inftyf(x_n)g(x_n) = fleft( lim_n to infty x_n right)gleft(lim_n to infty x_nright) = f(a)g(a)$$
$endgroup$
add a comment |
$begingroup$
The proof that the product of continuous functions is also continuous does not need to use square roots. You could add the condition that $0 < epsilon < 1$, in which case $epsilon^2 < epsilon$. Or you could use a convergent sequence $x_n$ such that $displaystyle lim_n mathop to infty x_n = a$, and show that
$$lim_n to inftyf(x_n)g(x_n) = fleft( lim_n to infty x_n right)gleft(lim_n to infty x_nright) = f(a)g(a)$$
$endgroup$
add a comment |
$begingroup$
The proof that the product of continuous functions is also continuous does not need to use square roots. You could add the condition that $0 < epsilon < 1$, in which case $epsilon^2 < epsilon$. Or you could use a convergent sequence $x_n$ such that $displaystyle lim_n mathop to infty x_n = a$, and show that
$$lim_n to inftyf(x_n)g(x_n) = fleft( lim_n to infty x_n right)gleft(lim_n to infty x_nright) = f(a)g(a)$$
$endgroup$
The proof that the product of continuous functions is also continuous does not need to use square roots. You could add the condition that $0 < epsilon < 1$, in which case $epsilon^2 < epsilon$. Or you could use a convergent sequence $x_n$ such that $displaystyle lim_n mathop to infty x_n = a$, and show that
$$lim_n to inftyf(x_n)g(x_n) = fleft( lim_n to infty x_n right)gleft(lim_n to infty x_nright) = f(a)g(a)$$
edited Jan 7 at 11:12
Mutantoe
604513
604513
answered Jan 7 at 9:14
gandalf61gandalf61
8,076625
8,076625
add a comment |
add a comment |
$begingroup$
You don't need existence of square root. You just need to have that $forall epsilon > 0 exists epsilon^prime > 0$ s. t. $epsilon^prime^2 < epsilon$. Then, you'll have $|(f(x) - K)(g(x)-L)| < epsilon^prime^2 < epsilon$ which gives you what you need anyway. You don't need the bound $epsilon$ to actually be reachable.
For $epsilon < 1$, $epsilon$ itself can serve as $epsilon^prime$.
$endgroup$
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
add a comment |
$begingroup$
You don't need existence of square root. You just need to have that $forall epsilon > 0 exists epsilon^prime > 0$ s. t. $epsilon^prime^2 < epsilon$. Then, you'll have $|(f(x) - K)(g(x)-L)| < epsilon^prime^2 < epsilon$ which gives you what you need anyway. You don't need the bound $epsilon$ to actually be reachable.
For $epsilon < 1$, $epsilon$ itself can serve as $epsilon^prime$.
$endgroup$
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
add a comment |
$begingroup$
You don't need existence of square root. You just need to have that $forall epsilon > 0 exists epsilon^prime > 0$ s. t. $epsilon^prime^2 < epsilon$. Then, you'll have $|(f(x) - K)(g(x)-L)| < epsilon^prime^2 < epsilon$ which gives you what you need anyway. You don't need the bound $epsilon$ to actually be reachable.
For $epsilon < 1$, $epsilon$ itself can serve as $epsilon^prime$.
$endgroup$
You don't need existence of square root. You just need to have that $forall epsilon > 0 exists epsilon^prime > 0$ s. t. $epsilon^prime^2 < epsilon$. Then, you'll have $|(f(x) - K)(g(x)-L)| < epsilon^prime^2 < epsilon$ which gives you what you need anyway. You don't need the bound $epsilon$ to actually be reachable.
For $epsilon < 1$, $epsilon$ itself can serve as $epsilon^prime$.
answered Jan 7 at 9:09
Todor MarkovTodor Markov
1,819410
1,819410
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
add a comment |
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
$begingroup$
IOW, you don't need the fact that $xmapsto x^2$ is continuous, just that it's monotonic for $x>0$.
$endgroup$
– leftaroundabout
Jan 7 at 20:59
add a comment |
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1
$begingroup$
You can use MathJax in a single formula - you don't have to write
$f(x)$<$g(x)$
and similar formulas. Also I see two different notations $lim_xto a$ and $Lim_xto a$ in the post - but without an explanation whether they are intended to represent two different things or they are both just different notations for the usual limit.$endgroup$
– Martin Sleziak
Jan 7 at 8:57
$begingroup$
Thank you. I have edited the question. The limit with lowercase L and uppercase L are not different, it was my mistake.
$endgroup$
– Steve
Jan 7 at 9:01
2
$begingroup$
Why don't you use the fact that $$|f(x) g(x) - LK|leq |g(x) ||f(x) - L|+|L||g(x) - K|$$ and show that each term on right can be made less than $epsilon /2$?
$endgroup$
– Paramanand Singh
Jan 7 at 9:10
1
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Use the fact that $g(x) to K$ and hence $|g(x) |<|K|+1$ and we can take $|f(x) - L|<epsilon /(2(|K|+1))$. This handles first term. In similar manner you can take $|g(x) - K|<epsilon/(2(|L|+1))$ to handle second term.
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– Paramanand Singh
Jan 7 at 9:29
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In general when you are dealing with $epsilon, delta$ proofs the expression based on these $epsilon, delta$ should not involve operations other than $+, -, times, /$.
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– Paramanand Singh
Jan 7 at 9:32