function limits and continuityLeft and right continuityUniform Continuity and DifferentiationRecasting the Definition of a Regulated functionContinuity understanding the definition and images and preimagesproblem about continuity and limitsConfusion over definition of continuity and limit of a functionQuestion about limit points in relation with continuity and functional limitsIf $lim_x to infty f'(x)$ is finite then $f(x)$ is uniformly continuousThe difference between continuity and uniform continuity.Real Analysis: Function ContinuityCheck the continuity of a function
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function limits and continuity
Left and right continuityUniform Continuity and DifferentiationRecasting the Definition of a Regulated functionContinuity understanding the definition and images and preimagesproblem about continuity and limitsConfusion over definition of continuity and limit of a functionQuestion about limit points in relation with continuity and functional limitsIf $lim_x to infty f'(x)$ is finite then $f(x)$ is uniformly continuousThe difference between continuity and uniform continuity.Real Analysis: Function ContinuityCheck the continuity of a function
$begingroup$
Suppose for a function, $f(x)$, which is defined on $[a,b]$ and for some $p$ in $[a,b]$. If $lim_x to p_+ f(x)$ and $lim_x to p_- f(x)$ are equal, say they are both $L$, am I therefore correct in concluding that $lim_x to p f(x)=L$. Also if $f(x)$ is continuous at $p$, is it true that then $f(p)=L$.
(Also if I would like to prove the second statement, do I just use the definition of limit and continuity to prove it?)
Much thanks in advance!
real-analysis
edited Mar 29 at 17:05
blub
3,167829
asked Mar 29 at 17:00
JustWanderingJustWandering
542
$endgroup$
add a comment |
$begingroup$
Suppose for a function, $f(x)$, which is defined on $[a,b]$ and for some $p$ in $[a,b]$. If $lim_x to p_+ f(x)$ and $lim_x to p_- f(x)$ are equal, say they are both $L$, am I therefore correct in concluding that $lim_x to p f(x)=L$. Also if $f(x)$ is continuous at $p$, is it true that then $f(p)=L$.
(Also if I would like to prove the second statement, do I just use the definition of limit and continuity to prove it?)
Much thanks in advance!
real-analysis
edited Mar 29 at 17:05
blub
3,167829
asked Mar 29 at 17:00
JustWanderingJustWandering
542
$endgroup$
2
$begingroup$
Possible duplicate of Left and right continuity
$endgroup$
– blub
Mar 29 at 17:07
add a comment |
$begingroup$
Suppose for a function, $f(x)$, which is defined on $[a,b]$ and for some $p$ in $[a,b]$. If $lim_x to p_+ f(x)$ and $lim_x to p_- f(x)$ are equal, say they are both $L$, am I therefore correct in concluding that $lim_x to p f(x)=L$. Also if $f(x)$ is continuous at $p$, is it true that then $f(p)=L$.
(Also if I would like to prove the second statement, do I just use the definition of limit and continuity to prove it?)
Much thanks in advance!
real-analysis
edited Mar 29 at 17:05
blub
3,167829
asked Mar 29 at 17:00
JustWanderingJustWandering
542
$endgroup$
Suppose for a function, $f(x)$, which is defined on $[a,b]$ and for some $p$ in $[a,b]$. If $lim_x to p_+ f(x)$ and $lim_x to p_- f(x)$ are equal, say they are both $L$, am I therefore correct in concluding that $lim_x to p f(x)=L$. Also if $f(x)$ is continuous at $p$, is it true that then $f(p)=L$.
(Also if I would like to prove the second statement, do I just use the definition of limit and continuity to prove it?)
Much thanks in advance!
real-analysis
real-analysis
edited Mar 29 at 17:05
blub
3,167829
asked Mar 29 at 17:00
JustWanderingJustWandering
542
edited Mar 29 at 17:05
blub
3,167829
asked Mar 29 at 17:00
JustWanderingJustWandering
542
edited Mar 29 at 17:05
blub
3,167829
edited Mar 29 at 17:05
blub
3,167829
edited Mar 29 at 17:05
blub
3,167829
3,167829
asked Mar 29 at 17:00
JustWanderingJustWandering
542
asked Mar 29 at 17:00
JustWanderingJustWandering
542
asked Mar 29 at 17:00
JustWanderingJustWandering
542
542
2
$begingroup$
Possible duplicate of Left and right continuity
$endgroup$
– blub
Mar 29 at 17:07
add a comment |
2
$begingroup$
Possible duplicate of Left and right continuity
$endgroup$
– blub
Mar 29 at 17:07
2
2
$begingroup$
Possible duplicate of Left and right continuity
$endgroup$
– blub
Mar 29 at 17:07
$begingroup$
Possible duplicate of Left and right continuity
$endgroup$
– blub
Mar 29 at 17:07
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Continuity of $f$ at $p$ is indeed achieved if and only if $lim_xto pf(x)=f(p)$. (It is implicit that the limit must exist.)
If you compare the definitions of a limit and of continuity, you will notice that they just differ in $L$ vs. $f(p)$.
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
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oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Continuity of $f$ at $p$ is indeed achieved if and only if $lim_xto pf(x)=f(p)$. (It is implicit that the limit must exist.)
If you compare the definitions of a limit and of continuity, you will notice that they just differ in $L$ vs. $f(p)$.
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
$endgroup$
add a comment |
$begingroup$
Continuity of $f$ at $p$ is indeed achieved if and only if $lim_xto pf(x)=f(p)$. (It is implicit that the limit must exist.)
If you compare the definitions of a limit and of continuity, you will notice that they just differ in $L$ vs. $f(p)$.
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
$endgroup$
add a comment |
$begingroup$
Continuity of $f$ at $p$ is indeed achieved if and only if $lim_xto pf(x)=f(p)$. (It is implicit that the limit must exist.)
If you compare the definitions of a limit and of continuity, you will notice that they just differ in $L$ vs. $f(p)$.
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
$endgroup$
Continuity of $f$ at $p$ is indeed achieved if and only if $lim_xto pf(x)=f(p)$. (It is implicit that the limit must exist.)
If you compare the definitions of a limit and of continuity, you will notice that they just differ in $L$ vs. $f(p)$.
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
answered Mar 29 at 17:11
Yves DaoustYves Daoust
132k676230
132k676230
add a comment |
add a comment |
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2
$begingroup$
Possible duplicate of Left and right continuity
$endgroup$
– blub
Mar 29 at 17:07