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Continuous differentiability of functions of one real variable
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Let $U$ be an open subset of $mathbb R$ and let $f:Utomathbb R$ be a differentiable function. Define the function $F:Utimes Utomathbb R$ by $$F(x,y)=begincases
dfracf(x)-f(y)x-y&textrm in case xne y\
f'(x)&textrm in case x=y.
endcases$$
Then I notice that $f'$ is continuous if and only if $F$ is continuous. So this gives a criterion for the continuous differentiability of functions of one real variable. Is there a related criterion for functions of several variables?
multivariable-calculus derivatives continuity
edited Jul 24 at 17:49
Math1000
18.4k31444
asked Jul 24 at 16:50
Peter Kropholler
111
add a comment |Â
up vote
0
down vote
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Let $U$ be an open subset of $mathbb R$ and let $f:Utomathbb R$ be a differentiable function. Define the function $F:Utimes Utomathbb R$ by $$F(x,y)=begincases
dfracf(x)-f(y)x-y&textrm in case xne y\
f'(x)&textrm in case x=y.
endcases$$
Then I notice that $f'$ is continuous if and only if $F$ is continuous. So this gives a criterion for the continuous differentiability of functions of one real variable. Is there a related criterion for functions of several variables?
multivariable-calculus derivatives continuity
edited Jul 24 at 17:49
Math1000
18.4k31444
asked Jul 24 at 16:50
Peter Kropholler
111
What exactly do you mean by $f'$ is continuous if $F$ is continuous? As $F(x,x) = f'(x)$, what exactly has your method going for it?
– Sudix
Jul 25 at 2:56
If $F$ is continuous then so is the composite $xmapsto(x,x)mapsto F(x,x)$ and hence $f'$ is continuous.
– Peter Kropholler
Jul 25 at 12:00
But that's essentially not a single bit easier than just checking for continuity on $f'$ itself
– Sudix
Jul 25 at 19:09
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $U$ be an open subset of $mathbb R$ and let $f:Utomathbb R$ be a differentiable function. Define the function $F:Utimes Utomathbb R$ by $$F(x,y)=begincases
dfracf(x)-f(y)x-y&textrm in case xne y\
f'(x)&textrm in case x=y.
endcases$$
Then I notice that $f'$ is continuous if and only if $F$ is continuous. So this gives a criterion for the continuous differentiability of functions of one real variable. Is there a related criterion for functions of several variables?
multivariable-calculus derivatives continuity
edited Jul 24 at 17:49
Math1000
18.4k31444
asked Jul 24 at 16:50
Peter Kropholler
111
Let $U$ be an open subset of $mathbb R$ and let $f:Utomathbb R$ be a differentiable function. Define the function $F:Utimes Utomathbb R$ by $$F(x,y)=begincases
dfracf(x)-f(y)x-y&textrm in case xne y\
f'(x)&textrm in case x=y.
endcases$$
Then I notice that $f'$ is continuous if and only if $F$ is continuous. So this gives a criterion for the continuous differentiability of functions of one real variable. Is there a related criterion for functions of several variables?
multivariable-calculus derivatives continuity
edited Jul 24 at 17:49
Math1000
18.4k31444
asked Jul 24 at 16:50
Peter Kropholler
111
edited Jul 24 at 17:49
Math1000
18.4k31444
edited Jul 24 at 17:49
Math1000
18.4k31444
edited Jul 24 at 17:49
Math1000
18.4k31444
18.4k31444
asked Jul 24 at 16:50
Peter Kropholler
111
asked Jul 24 at 16:50
Peter Kropholler
111
asked Jul 24 at 16:50
Peter Kropholler
111
111
What exactly do you mean by $f'$ is continuous if $F$ is continuous? As $F(x,x) = f'(x)$, what exactly has your method going for it?
– Sudix
Jul 25 at 2:56
If $F$ is continuous then so is the composite $xmapsto(x,x)mapsto F(x,x)$ and hence $f'$ is continuous.
– Peter Kropholler
Jul 25 at 12:00
But that's essentially not a single bit easier than just checking for continuity on $f'$ itself
– Sudix
Jul 25 at 19:09
add a comment |Â
What exactly do you mean by $f'$ is continuous if $F$ is continuous? As $F(x,x) = f'(x)$, what exactly has your method going for it?
– Sudix
Jul 25 at 2:56
If $F$ is continuous then so is the composite $xmapsto(x,x)mapsto F(x,x)$ and hence $f'$ is continuous.
– Peter Kropholler
Jul 25 at 12:00
But that's essentially not a single bit easier than just checking for continuity on $f'$ itself
– Sudix
Jul 25 at 19:09
What exactly do you mean by $f'$ is continuous if $F$ is continuous? As $F(x,x) = f'(x)$, what exactly has your method going for it?
– Sudix
Jul 25 at 2:56
What exactly do you mean by $f'$ is continuous if $F$ is continuous? As $F(x,x) = f'(x)$, what exactly has your method going for it?
– Sudix
Jul 25 at 2:56
If $F$ is continuous then so is the composite $xmapsto(x,x)mapsto F(x,x)$ and hence $f'$ is continuous.
– Peter Kropholler
Jul 25 at 12:00
If $F$ is continuous then so is the composite $xmapsto(x,x)mapsto F(x,x)$ and hence $f'$ is continuous.
– Peter Kropholler
Jul 25 at 12:00
But that's essentially not a single bit easier than just checking for continuity on $f'$ itself
– Sudix
Jul 25 at 19:09
But that's essentially not a single bit easier than just checking for continuity on $f'$ itself
– Sudix
Jul 25 at 19:09
add a comment |Â
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What exactly do you mean by $f'$ is continuous if $F$ is continuous? As $F(x,x) = f'(x)$, what exactly has your method going for it?
– Sudix
Jul 25 at 2:56
If $F$ is continuous then so is the composite $xmapsto(x,x)mapsto F(x,x)$ and hence $f'$ is continuous.
– Peter Kropholler
Jul 25 at 12:00
But that's essentially not a single bit easier than just checking for continuity on $f'$ itself
– Sudix
Jul 25 at 19:09