Continuity of polynomials The 2019 Stack Overflow Developer Survey Results Are InHow can I prove that a polynomial with degree $n$ is continuous everywhere in $mathbbR$ using definitions?Commutative Diagrams and PolynomialsTo prove given $ r cdot f_1+f_2cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ isIs $r_1 cdot f_1 + r_2 cdot f_2 $ uniformly distributed?Observation about Polynomials AdditionHow to make addition of two polynomials have no integer root.Decomposition of the space of polynomialsA property holds generically for polynomials - meaning?If $lim_hto 0 frac f_1(a+ h) - 2f_1(a) + f_1(a-h) h^2$ is constant then is $f_1$ a quadratic function?Algebraic independence of certain values implies algebraic independence of functions?Algebraic structure of set of polynomials passing through points
What are the motivations for publishing new editions of an existing textbook, beyond new discoveries in a field?
What does Linus Torvalds mean when he says that Git "never ever" tracks a file?
Unbreakable Formation vs. Cry of the Carnarium
Does light intensity oscillate really fast since it is a wave?
How can I fix this gap between bookcases I made?
How to create dashed lines/arrows in Illustrator
What is the best strategy for white in this position?
Inversion Puzzle
"To split hairs" vs "To be pedantic"
If the Wish spell is used to duplicate the effect of Simulacrum, are existing duplicates destroyed?
Is this food a bread or a loaf?
Return to UK after being refused
Is an up-to-date browser secure on an out-of-date OS?
Limit to 0 ambiguity
Patience, young "Padovan"
Why is the maximum length of OpenWrt’s root password 8 characters?
Why do some words that are not inflected have an umlaut?
Where does the "burst of radiance" from Holy Weapon originate?
What is the steepest angle that a canal can be traversable without locks?
What is the use of option -o in the useradd command?
How to change the limits of integration
How come people say “Would of”?
CiviEvent: Public link for events of a specific type
Dual Citizen. Exited the US on Italian passport recently
Continuity of polynomials
The 2019 Stack Overflow Developer Survey Results Are InHow can I prove that a polynomial with degree $n$ is continuous everywhere in $mathbbR$ using definitions?Commutative Diagrams and PolynomialsTo prove given $ r cdot f_1+f_2cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ isIs $r_1 cdot f_1 + r_2 cdot f_2 $ uniformly distributed?Observation about Polynomials AdditionHow to make addition of two polynomials have no integer root.Decomposition of the space of polynomialsA property holds generically for polynomials - meaning?If $lim_hto 0 frac f_1(a+ h) - 2f_1(a) + f_1(a-h) h^2$ is constant then is $f_1$ a quadratic function?Algebraic independence of certain values implies algebraic independence of functions?Algebraic structure of set of polynomials passing through points
$begingroup$
I've seen an answer (How can I prove that a polynomial with degree $n$ is continuous everywhere in $mathbbR$ using definitions?) on the question to prove that all polynomials are continuous, so I tried to follow his steps. Could you please tell me if that is alright and if not tell me what I could consider? Also the 1,2, and 4 are fairly obvious(from my lecture notes) so I did not write them out. P.S How can I (start) prove that if r=p/q is a ratio of two polynomials then it is continuous at every point of R where q≠0.
1) $f(x)=x$ is continuous everywhere
2) If $f(x)$ and $g(x)$ are continuous in $D$ then $f(x)cdot g(x)$ in continuous on $D$.
3) Using $2$ and $1$ show that $x^n$ is continuous for every $n in mathbb N$
4) If $f(x)$ and $g(x)$ are continuous on $D$ then $f(x)+g(x)$ is continous on $D$
5) Now use $3$ and $4$.
So here is how I followed the steps:
1) elementary proof
2) algebraic property of coninuous functions proof
3) Proof: If $f_1(x)=x$ and $f_2(x)=x$ are both continuous on $D$ then from 2) we know that $f_1(x)cdot f_2(x)=x^2$ is continuous. Suppose now that $f_1(x)=x$ $f_2(x)=x^2$ $f_3(x)=x^3 ... f_m(x)=x^m$. Hence we can conclude that $f_1(x)cdot ....f_m(x)=x^n$ is continuous on $D$, for $n=m(m+1)/2$.
4) algebraic property of continuous functions
5) proof: from $3)$ we know that $x^n$ is contionuous on $D$ and hence if we add a polynomial of degree $n-1$ or smaller then by $4)$ we can conclude that $x^n+x^n-1+...+1$ is continuous.
Hence all polynomials are continuous.
real-analysis functions polynomials
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
$endgroup$
|
show 2 more comments
$begingroup$
I've seen an answer (How can I prove that a polynomial with degree $n$ is continuous everywhere in $mathbbR$ using definitions?) on the question to prove that all polynomials are continuous, so I tried to follow his steps. Could you please tell me if that is alright and if not tell me what I could consider? Also the 1,2, and 4 are fairly obvious(from my lecture notes) so I did not write them out. P.S How can I (start) prove that if r=p/q is a ratio of two polynomials then it is continuous at every point of R where q≠0.
1) $f(x)=x$ is continuous everywhere
2) If $f(x)$ and $g(x)$ are continuous in $D$ then $f(x)cdot g(x)$ in continuous on $D$.
3) Using $2$ and $1$ show that $x^n$ is continuous for every $n in mathbb N$
4) If $f(x)$ and $g(x)$ are continuous on $D$ then $f(x)+g(x)$ is continous on $D$
5) Now use $3$ and $4$.
So here is how I followed the steps:
1) elementary proof
2) algebraic property of coninuous functions proof
3) Proof: If $f_1(x)=x$ and $f_2(x)=x$ are both continuous on $D$ then from 2) we know that $f_1(x)cdot f_2(x)=x^2$ is continuous. Suppose now that $f_1(x)=x$ $f_2(x)=x^2$ $f_3(x)=x^3 ... f_m(x)=x^m$. Hence we can conclude that $f_1(x)cdot ....f_m(x)=x^n$ is continuous on $D$, for $n=m(m+1)/2$.
4) algebraic property of continuous functions
5) proof: from $3)$ we know that $x^n$ is contionuous on $D$ and hence if we add a polynomial of degree $n-1$ or smaller then by $4)$ we can conclude that $x^n+x^n-1+...+1$ is continuous.
Hence all polynomials are continuous.
real-analysis functions polynomials
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
$endgroup$
$begingroup$
There is one little matter with 5, since not every polynomial are in form of $x^n+x^n-1+...+1$, should prove for any $a_nx_n+a_n-1x^n-1+...+a_0$.
$endgroup$
– L KM
Mar 30 at 12:51
$begingroup$
For the second question regarding rational function, if $q(x)$ is not $0$ in some point $c$, then it is nonzero in a neighbourhood of $c$, then just apply $epsilon - delta$ defintion as usual.
$endgroup$
– L KM
Mar 30 at 12:53
$begingroup$
Also, in step 3, $n=m$, not $m(m+1)/2$.
$endgroup$
– L KM
Mar 30 at 12:58
$begingroup$
Well on step 3 I said that n=m(m+1)/2 since x.x^2.x^3...x^m=x^n, isnt that right? or that's an unnecessary thing?
$endgroup$
– The Poor Jew
Mar 30 at 13:04
$begingroup$
But your definition seems to be $f_1(x)=f_2(x)=...=f_m(x)=x$
$endgroup$
– L KM
Mar 30 at 13:06
|
show 2 more comments
$begingroup$
I've seen an answer (How can I prove that a polynomial with degree $n$ is continuous everywhere in $mathbbR$ using definitions?) on the question to prove that all polynomials are continuous, so I tried to follow his steps. Could you please tell me if that is alright and if not tell me what I could consider? Also the 1,2, and 4 are fairly obvious(from my lecture notes) so I did not write them out. P.S How can I (start) prove that if r=p/q is a ratio of two polynomials then it is continuous at every point of R where q≠0.
1) $f(x)=x$ is continuous everywhere
2) If $f(x)$ and $g(x)$ are continuous in $D$ then $f(x)cdot g(x)$ in continuous on $D$.
3) Using $2$ and $1$ show that $x^n$ is continuous for every $n in mathbb N$
4) If $f(x)$ and $g(x)$ are continuous on $D$ then $f(x)+g(x)$ is continous on $D$
5) Now use $3$ and $4$.
So here is how I followed the steps:
1) elementary proof
2) algebraic property of coninuous functions proof
3) Proof: If $f_1(x)=x$ and $f_2(x)=x$ are both continuous on $D$ then from 2) we know that $f_1(x)cdot f_2(x)=x^2$ is continuous. Suppose now that $f_1(x)=x$ $f_2(x)=x^2$ $f_3(x)=x^3 ... f_m(x)=x^m$. Hence we can conclude that $f_1(x)cdot ....f_m(x)=x^n$ is continuous on $D$, for $n=m(m+1)/2$.
4) algebraic property of continuous functions
5) proof: from $3)$ we know that $x^n$ is contionuous on $D$ and hence if we add a polynomial of degree $n-1$ or smaller then by $4)$ we can conclude that $x^n+x^n-1+...+1$ is continuous.
Hence all polynomials are continuous.
real-analysis functions polynomials
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
$endgroup$
I've seen an answer (How can I prove that a polynomial with degree $n$ is continuous everywhere in $mathbbR$ using definitions?) on the question to prove that all polynomials are continuous, so I tried to follow his steps. Could you please tell me if that is alright and if not tell me what I could consider? Also the 1,2, and 4 are fairly obvious(from my lecture notes) so I did not write them out. P.S How can I (start) prove that if r=p/q is a ratio of two polynomials then it is continuous at every point of R where q≠0.
1) $f(x)=x$ is continuous everywhere
2) If $f(x)$ and $g(x)$ are continuous in $D$ then $f(x)cdot g(x)$ in continuous on $D$.
3) Using $2$ and $1$ show that $x^n$ is continuous for every $n in mathbb N$
4) If $f(x)$ and $g(x)$ are continuous on $D$ then $f(x)+g(x)$ is continous on $D$
5) Now use $3$ and $4$.
So here is how I followed the steps:
1) elementary proof
2) algebraic property of coninuous functions proof
3) Proof: If $f_1(x)=x$ and $f_2(x)=x$ are both continuous on $D$ then from 2) we know that $f_1(x)cdot f_2(x)=x^2$ is continuous. Suppose now that $f_1(x)=x$ $f_2(x)=x^2$ $f_3(x)=x^3 ... f_m(x)=x^m$. Hence we can conclude that $f_1(x)cdot ....f_m(x)=x^n$ is continuous on $D$, for $n=m(m+1)/2$.
4) algebraic property of continuous functions
5) proof: from $3)$ we know that $x^n$ is contionuous on $D$ and hence if we add a polynomial of degree $n-1$ or smaller then by $4)$ we can conclude that $x^n+x^n-1+...+1$ is continuous.
Hence all polynomials are continuous.
real-analysis functions polynomials
real-analysis functions polynomials
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
edited Mar 30 at 14:03
The Poor Jew
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
asked Mar 30 at 12:44
The Poor JewThe Poor Jew
747
747
$begingroup$
There is one little matter with 5, since not every polynomial are in form of $x^n+x^n-1+...+1$, should prove for any $a_nx_n+a_n-1x^n-1+...+a_0$.
$endgroup$
– L KM
Mar 30 at 12:51
$begingroup$
For the second question regarding rational function, if $q(x)$ is not $0$ in some point $c$, then it is nonzero in a neighbourhood of $c$, then just apply $epsilon - delta$ defintion as usual.
$endgroup$
– L KM
Mar 30 at 12:53
$begingroup$
Also, in step 3, $n=m$, not $m(m+1)/2$.
$endgroup$
– L KM
Mar 30 at 12:58
$begingroup$
Well on step 3 I said that n=m(m+1)/2 since x.x^2.x^3...x^m=x^n, isnt that right? or that's an unnecessary thing?
$endgroup$
– The Poor Jew
Mar 30 at 13:04
$begingroup$
But your definition seems to be $f_1(x)=f_2(x)=...=f_m(x)=x$
$endgroup$
– L KM
Mar 30 at 13:06
|
show 2 more comments
$begingroup$
There is one little matter with 5, since not every polynomial are in form of $x^n+x^n-1+...+1$, should prove for any $a_nx_n+a_n-1x^n-1+...+a_0$.
$endgroup$
– L KM
Mar 30 at 12:51
$begingroup$
For the second question regarding rational function, if $q(x)$ is not $0$ in some point $c$, then it is nonzero in a neighbourhood of $c$, then just apply $epsilon - delta$ defintion as usual.
$endgroup$
– L KM
Mar 30 at 12:53
$begingroup$
Also, in step 3, $n=m$, not $m(m+1)/2$.
$endgroup$
– L KM
Mar 30 at 12:58
$begingroup$
Well on step 3 I said that n=m(m+1)/2 since x.x^2.x^3...x^m=x^n, isnt that right? or that's an unnecessary thing?
$endgroup$
– The Poor Jew
Mar 30 at 13:04
$begingroup$
But your definition seems to be $f_1(x)=f_2(x)=...=f_m(x)=x$
$endgroup$
– L KM
Mar 30 at 13:06
$begingroup$
There is one little matter with 5, since not every polynomial are in form of $x^n+x^n-1+...+1$, should prove for any $a_nx_n+a_n-1x^n-1+...+a_0$.
$endgroup$
– L KM
Mar 30 at 12:51
$begingroup$
There is one little matter with 5, since not every polynomial are in form of $x^n+x^n-1+...+1$, should prove for any $a_nx_n+a_n-1x^n-1+...+a_0$.
$endgroup$
– L KM
Mar 30 at 12:51
$begingroup$
For the second question regarding rational function, if $q(x)$ is not $0$ in some point $c$, then it is nonzero in a neighbourhood of $c$, then just apply $epsilon - delta$ defintion as usual.
$endgroup$
– L KM
Mar 30 at 12:53
$begingroup$
For the second question regarding rational function, if $q(x)$ is not $0$ in some point $c$, then it is nonzero in a neighbourhood of $c$, then just apply $epsilon - delta$ defintion as usual.
$endgroup$
– L KM
Mar 30 at 12:53
$begingroup$
Also, in step 3, $n=m$, not $m(m+1)/2$.
$endgroup$
– L KM
Mar 30 at 12:58
$begingroup$
Also, in step 3, $n=m$, not $m(m+1)/2$.
$endgroup$
– L KM
Mar 30 at 12:58
$begingroup$
Well on step 3 I said that n=m(m+1)/2 since x.x^2.x^3...x^m=x^n, isnt that right? or that's an unnecessary thing?
$endgroup$
– The Poor Jew
Mar 30 at 13:04
$begingroup$
Well on step 3 I said that n=m(m+1)/2 since x.x^2.x^3...x^m=x^n, isnt that right? or that's an unnecessary thing?
$endgroup$
– The Poor Jew
Mar 30 at 13:04
$begingroup$
But your definition seems to be $f_1(x)=f_2(x)=...=f_m(x)=x$
$endgroup$
– L KM
Mar 30 at 13:06
$begingroup$
But your definition seems to be $f_1(x)=f_2(x)=...=f_m(x)=x$
$endgroup$
– L KM
Mar 30 at 13:06
|
show 2 more comments
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3168257%2fcontinuity-of-polynomials%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3168257%2fcontinuity-of-polynomials%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
There is one little matter with 5, since not every polynomial are in form of $x^n+x^n-1+...+1$, should prove for any $a_nx_n+a_n-1x^n-1+...+a_0$.
$endgroup$
– L KM
Mar 30 at 12:51
$begingroup$
For the second question regarding rational function, if $q(x)$ is not $0$ in some point $c$, then it is nonzero in a neighbourhood of $c$, then just apply $epsilon - delta$ defintion as usual.
$endgroup$
– L KM
Mar 30 at 12:53
$begingroup$
Also, in step 3, $n=m$, not $m(m+1)/2$.
$endgroup$
– L KM
Mar 30 at 12:58
$begingroup$
Well on step 3 I said that n=m(m+1)/2 since x.x^2.x^3...x^m=x^n, isnt that right? or that's an unnecessary thing?
$endgroup$
– The Poor Jew
Mar 30 at 13:04
$begingroup$
But your definition seems to be $f_1(x)=f_2(x)=...=f_m(x)=x$
$endgroup$
– L KM
Mar 30 at 13:06