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Can we allow $f$ to be undefined at finitely many points in $(a,b)$ when formulating $int_a^b f(x) dx=F(b)-F(a)$, ($F$ is the antiderivative of $f$)



The Next CEO of Stack OverflowChanging one point does not change the Riemann integralIs there a definite integral for which the Riemann sum can be calculated but for which there is no closed-form antiderivative?When can the order of limit and integral be exchanged?Evaluation of improper integralsAre all antiderivatives of a function an area or a (difference of areas) function of its derivative?How to establish the equivalence of these two statements about integrals of step functions?How is $int f(x) dx$ different from $int_0^x f(t)dt$?Antiderivative and definite integralIf functions agree at all but finitely many points then the integrals are the sameWhy is it that, when using the u-substitution rule, you apply the the function u to the upper and lower limits?Various Questions On Integration.










4












$begingroup$


Let $f$ be a real-valued function on a closed interval $[a,b]$ undefined only at finite points in $(a,b)$. Let $F$ be antiderivative of $f$. Then:



$$int_a^b f(x) dx=F(b)-F(a)$$



Is the theorem true? How shall we prove it?




I ask this because I have read that function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral.











share|cite|improve this question











$endgroup$











  • $begingroup$
    @Asaf Karagila: Thanks for the edit.
    $endgroup$
    – Joe
    Mar 18 at 11:27










  • $begingroup$
    You're welcome. Titles have 150 characters, and you should use them whenever possible to give a clear description of your question.
    $endgroup$
    – Asaf Karagila
    Mar 18 at 11:28










  • $begingroup$
    Understood.......................
    $endgroup$
    – Joe
    Mar 18 at 11:29















4












$begingroup$


Let $f$ be a real-valued function on a closed interval $[a,b]$ undefined only at finite points in $(a,b)$. Let $F$ be antiderivative of $f$. Then:



$$int_a^b f(x) dx=F(b)-F(a)$$



Is the theorem true? How shall we prove it?




I ask this because I have read that function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral.











share|cite|improve this question











$endgroup$











  • $begingroup$
    @Asaf Karagila: Thanks for the edit.
    $endgroup$
    – Joe
    Mar 18 at 11:27










  • $begingroup$
    You're welcome. Titles have 150 characters, and you should use them whenever possible to give a clear description of your question.
    $endgroup$
    – Asaf Karagila
    Mar 18 at 11:28










  • $begingroup$
    Understood.......................
    $endgroup$
    – Joe
    Mar 18 at 11:29













4












4








4





$begingroup$


Let $f$ be a real-valued function on a closed interval $[a,b]$ undefined only at finite points in $(a,b)$. Let $F$ be antiderivative of $f$. Then:



$$int_a^b f(x) dx=F(b)-F(a)$$



Is the theorem true? How shall we prove it?




I ask this because I have read that function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral.











share|cite|improve this question











$endgroup$




Let $f$ be a real-valued function on a closed interval $[a,b]$ undefined only at finite points in $(a,b)$. Let $F$ be antiderivative of $f$. Then:



$$int_a^b f(x) dx=F(b)-F(a)$$



Is the theorem true? How shall we prove it?




I ask this because I have read that function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral.








calculus integration limits functions definite-integrals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 18 at 11:21









Asaf Karagila

307k33439771




307k33439771










asked Mar 18 at 7:06









JoeJoe

313214




313214











  • $begingroup$
    @Asaf Karagila: Thanks for the edit.
    $endgroup$
    – Joe
    Mar 18 at 11:27










  • $begingroup$
    You're welcome. Titles have 150 characters, and you should use them whenever possible to give a clear description of your question.
    $endgroup$
    – Asaf Karagila
    Mar 18 at 11:28










  • $begingroup$
    Understood.......................
    $endgroup$
    – Joe
    Mar 18 at 11:29
















  • $begingroup$
    @Asaf Karagila: Thanks for the edit.
    $endgroup$
    – Joe
    Mar 18 at 11:27










  • $begingroup$
    You're welcome. Titles have 150 characters, and you should use them whenever possible to give a clear description of your question.
    $endgroup$
    – Asaf Karagila
    Mar 18 at 11:28










  • $begingroup$
    Understood.......................
    $endgroup$
    – Joe
    Mar 18 at 11:29















$begingroup$
@Asaf Karagila: Thanks for the edit.
$endgroup$
– Joe
Mar 18 at 11:27




$begingroup$
@Asaf Karagila: Thanks for the edit.
$endgroup$
– Joe
Mar 18 at 11:27












$begingroup$
You're welcome. Titles have 150 characters, and you should use them whenever possible to give a clear description of your question.
$endgroup$
– Asaf Karagila
Mar 18 at 11:28




$begingroup$
You're welcome. Titles have 150 characters, and you should use them whenever possible to give a clear description of your question.
$endgroup$
– Asaf Karagila
Mar 18 at 11:28












$begingroup$
Understood.......................
$endgroup$
– Joe
Mar 18 at 11:29




$begingroup$
Understood.......................
$endgroup$
– Joe
Mar 18 at 11:29










1 Answer
1






active

oldest

votes


















6












$begingroup$

That statement doesn't make sense, since $f$ may well not be integrable on $[a,b]$ then. And if it is, the statment is not necessarily true. Take, for instance, the null function $f$ on $[0,2]setminus1$. Then$$beginarrayrcccFcolon&[0,2]setminus1&longrightarrow&mathbb R\&x&mapsto&begincases0&text if xin[0,1)\1&text if xin(1,2]endcasesendarray$$is an antiderivative of $f$, but$$int_0^2f(x),mathrm dx=0neq1=F(2)-F(0).$$






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
    $endgroup$
    – Joe
    Mar 18 at 7:16










  • $begingroup$
    It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:25










  • $begingroup$
    Yes exactly that... Why is it so?
    $endgroup$
    – Joe
    Mar 18 at 7:35










  • $begingroup$
    Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:48











  • $begingroup$
    @Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
    $endgroup$
    – lEm
    Mar 18 at 7:50











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









6












$begingroup$

That statement doesn't make sense, since $f$ may well not be integrable on $[a,b]$ then. And if it is, the statment is not necessarily true. Take, for instance, the null function $f$ on $[0,2]setminus1$. Then$$beginarrayrcccFcolon&[0,2]setminus1&longrightarrow&mathbb R\&x&mapsto&begincases0&text if xin[0,1)\1&text if xin(1,2]endcasesendarray$$is an antiderivative of $f$, but$$int_0^2f(x),mathrm dx=0neq1=F(2)-F(0).$$






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
    $endgroup$
    – Joe
    Mar 18 at 7:16










  • $begingroup$
    It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:25










  • $begingroup$
    Yes exactly that... Why is it so?
    $endgroup$
    – Joe
    Mar 18 at 7:35










  • $begingroup$
    Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:48











  • $begingroup$
    @Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
    $endgroup$
    – lEm
    Mar 18 at 7:50















6












$begingroup$

That statement doesn't make sense, since $f$ may well not be integrable on $[a,b]$ then. And if it is, the statment is not necessarily true. Take, for instance, the null function $f$ on $[0,2]setminus1$. Then$$beginarrayrcccFcolon&[0,2]setminus1&longrightarrow&mathbb R\&x&mapsto&begincases0&text if xin[0,1)\1&text if xin(1,2]endcasesendarray$$is an antiderivative of $f$, but$$int_0^2f(x),mathrm dx=0neq1=F(2)-F(0).$$






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
    $endgroup$
    – Joe
    Mar 18 at 7:16










  • $begingroup$
    It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:25










  • $begingroup$
    Yes exactly that... Why is it so?
    $endgroup$
    – Joe
    Mar 18 at 7:35










  • $begingroup$
    Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:48











  • $begingroup$
    @Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
    $endgroup$
    – lEm
    Mar 18 at 7:50













6












6








6





$begingroup$

That statement doesn't make sense, since $f$ may well not be integrable on $[a,b]$ then. And if it is, the statment is not necessarily true. Take, for instance, the null function $f$ on $[0,2]setminus1$. Then$$beginarrayrcccFcolon&[0,2]setminus1&longrightarrow&mathbb R\&x&mapsto&begincases0&text if xin[0,1)\1&text if xin(1,2]endcasesendarray$$is an antiderivative of $f$, but$$int_0^2f(x),mathrm dx=0neq1=F(2)-F(0).$$






share|cite|improve this answer









$endgroup$



That statement doesn't make sense, since $f$ may well not be integrable on $[a,b]$ then. And if it is, the statment is not necessarily true. Take, for instance, the null function $f$ on $[0,2]setminus1$. Then$$beginarrayrcccFcolon&[0,2]setminus1&longrightarrow&mathbb R\&x&mapsto&begincases0&text if xin[0,1)\1&text if xin(1,2]endcasesendarray$$is an antiderivative of $f$, but$$int_0^2f(x),mathrm dx=0neq1=F(2)-F(0).$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 18 at 7:12









José Carlos SantosJosé Carlos Santos

171k23132240




171k23132240











  • $begingroup$
    Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
    $endgroup$
    – Joe
    Mar 18 at 7:16










  • $begingroup$
    It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:25










  • $begingroup$
    Yes exactly that... Why is it so?
    $endgroup$
    – Joe
    Mar 18 at 7:35










  • $begingroup$
    Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:48











  • $begingroup$
    @Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
    $endgroup$
    – lEm
    Mar 18 at 7:50
















  • $begingroup$
    Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
    $endgroup$
    – Joe
    Mar 18 at 7:16










  • $begingroup$
    It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:25










  • $begingroup$
    Yes exactly that... Why is it so?
    $endgroup$
    – Joe
    Mar 18 at 7:35










  • $begingroup$
    Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
    $endgroup$
    – José Carlos Santos
    Mar 18 at 7:48











  • $begingroup$
    @Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
    $endgroup$
    – lEm
    Mar 18 at 7:50















$begingroup$
Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
$endgroup$
– Joe
Mar 18 at 7:16




$begingroup$
Ok... But what about the statement "function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral."
$endgroup$
– Joe
Mar 18 at 7:16












$begingroup$
It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
$endgroup$
– José Carlos Santos
Mar 18 at 7:25




$begingroup$
It's rather vague. I suppose that the idea was that if the domain of $f$ is something like $(a,b)setminusc,d$, with $c<d$, we could use the standard method of antiderivatives to compute $int_a^cf(x),mathrm dx$, $int_c^df(x),mathrm dx$, and $int_d^bf(x),mathrm dx$.
$endgroup$
– José Carlos Santos
Mar 18 at 7:25












$begingroup$
Yes exactly that... Why is it so?
$endgroup$
– Joe
Mar 18 at 7:35




$begingroup$
Yes exactly that... Why is it so?
$endgroup$
– Joe
Mar 18 at 7:35












$begingroup$
Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
$endgroup$
– José Carlos Santos
Mar 18 at 7:48





$begingroup$
Because, if $f$ is integrable,$$int_a^bf(x),mathrm dx=int_a^cf(x),mathrm dx+int_c^df(x),mathrm dx+int_d^af(x),mathrm dx.$$
$endgroup$
– José Carlos Santos
Mar 18 at 7:48













$begingroup$
@Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
$endgroup$
– lEm
Mar 18 at 7:50




$begingroup$
@Joe To begin answering that question, you can't integrate a function that isn't even well-defined! A better statement would be, if $f$ and $g$ are integrable functions so that their values differ at finitely many points, then they have the same integrals. Now if you have learned about Riemann sum, the main idea is that you can always take a small enough partition so that the intervals containing those finitely many points have lengths as small as you wish.
$endgroup$
– lEm
Mar 18 at 7:50

















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Lowndes Grove History Architecture References Navigation menu32°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661132°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661178002500"National Register Information System"Historic houses of South Carolina"Lowndes Grove""+32° 48' 6.00", −79° 57' 58.00""Lowndes Grove, Charleston County (260 St. Margaret St., Charleston)""Lowndes Grove"The Charleston ExpositionIt Happened in South Carolina"Lowndes Grove (House), Saint Margaret Street & Sixth Avenue, Charleston, Charleston County, SC(Photographs)"Plantations of the Carolina Low Countrye