A global, elementary, implicit function theory
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When I was a student, my instructor proposed me a nice and elementary implicit function theory that I can summarize as follows: suppose $f colon I times mathbb{R} to mathbb{R}$ is a sufficiently smooth function defined on a cylinder $I times mathbb{R} subset mathbb{R}^2$, where $I$ is an interval. If $lim_{y to -infty} f(x,y)<0$, $lim_{y to +infty} f(x,y)>0$ and $f(x,cdot)$ is strictly increasing for each $x in J$, then $f(x,y)=0$ defines a unique function $y colon I to mathbb{R}$. The function $y$ is as regular as $f$ is.
Of course the proof is almost trivial. My question is where I can find a reference in english, since I would like to use this result without proof in a paper. I known of a textbook in italian that presents this result, but italian is not read by so many people around the world.
real-analysis multivariable-calculus reference-request
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
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add a comment |
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When I was a student, my instructor proposed me a nice and elementary implicit function theory that I can summarize as follows: suppose $f colon I times mathbb{R} to mathbb{R}$ is a sufficiently smooth function defined on a cylinder $I times mathbb{R} subset mathbb{R}^2$, where $I$ is an interval. If $lim_{y to -infty} f(x,y)<0$, $lim_{y to +infty} f(x,y)>0$ and $f(x,cdot)$ is strictly increasing for each $x in J$, then $f(x,y)=0$ defines a unique function $y colon I to mathbb{R}$. The function $y$ is as regular as $f$ is.
Of course the proof is almost trivial. My question is where I can find a reference in english, since I would like to use this result without proof in a paper. I known of a textbook in italian that presents this result, but italian is not read by so many people around the world.
real-analysis multivariable-calculus reference-request
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
$endgroup$
add a comment |
$begingroup$
When I was a student, my instructor proposed me a nice and elementary implicit function theory that I can summarize as follows: suppose $f colon I times mathbb{R} to mathbb{R}$ is a sufficiently smooth function defined on a cylinder $I times mathbb{R} subset mathbb{R}^2$, where $I$ is an interval. If $lim_{y to -infty} f(x,y)<0$, $lim_{y to +infty} f(x,y)>0$ and $f(x,cdot)$ is strictly increasing for each $x in J$, then $f(x,y)=0$ defines a unique function $y colon I to mathbb{R}$. The function $y$ is as regular as $f$ is.
Of course the proof is almost trivial. My question is where I can find a reference in english, since I would like to use this result without proof in a paper. I known of a textbook in italian that presents this result, but italian is not read by so many people around the world.
real-analysis multivariable-calculus reference-request
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
$endgroup$
When I was a student, my instructor proposed me a nice and elementary implicit function theory that I can summarize as follows: suppose $f colon I times mathbb{R} to mathbb{R}$ is a sufficiently smooth function defined on a cylinder $I times mathbb{R} subset mathbb{R}^2$, where $I$ is an interval. If $lim_{y to -infty} f(x,y)<0$, $lim_{y to +infty} f(x,y)>0$ and $f(x,cdot)$ is strictly increasing for each $x in J$, then $f(x,y)=0$ defines a unique function $y colon I to mathbb{R}$. The function $y$ is as regular as $f$ is.
Of course the proof is almost trivial. My question is where I can find a reference in english, since I would like to use this result without proof in a paper. I known of a textbook in italian that presents this result, but italian is not read by so many people around the world.
real-analysis multivariable-calculus reference-request
real-analysis multivariable-calculus reference-request
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
asked Jan 9 at 10:48
SiminoreSiminore
30.6k33569
30.6k33569
add a comment |
add a comment |
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