21.G. Let f be a function on R to R? which is differentiable on a neighborhood of a point c and such that Df (c) has an inverse. Then is it true that f has an inverse on a neighborhood of c?

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 24E
icon
Related questions
icon
Concept explainers
Question

please help with exercise 21.G

detailed solution for you to understand

21.G. Let f be a function on R to R? which is differentiable on a neighborhood
of a point c and such that Df (c) has an inverse. Then is it true that f has an
inverse on a neighborhood of c?
21.H. Let f be a funetion on Rº to RP. If f is differentiable at c and has a
differentiable inverse, then is it true that Df(c) is one-one?
21.1. Suppose that f is differentiable on a neighborhood of a point c and that
if e > 0 then there exists 6(e) > 0 such that if |r - c| < ô(e), then |Df(x) (2) –
Df(c) (2) < elz| for all z in RP. Prove that the partial derivatives of f exist and
are continuous at c.
21.J. Suppose that Lo is a one-one linear function on Rr to R. Show that
there exists a positive number a such that if L is a linear function on Rr to R
satisfying
Transcribed Image Text:21.G. Let f be a function on R to R? which is differentiable on a neighborhood of a point c and such that Df (c) has an inverse. Then is it true that f has an inverse on a neighborhood of c? 21.H. Let f be a funetion on Rº to RP. If f is differentiable at c and has a differentiable inverse, then is it true that Df(c) is one-one? 21.1. Suppose that f is differentiable on a neighborhood of a point c and that if e > 0 then there exists 6(e) > 0 such that if |r - c| < ô(e), then |Df(x) (2) – Df(c) (2) < elz| for all z in RP. Prove that the partial derivatives of f exist and are continuous at c. 21.J. Suppose that Lo is a one-one linear function on Rr to R. Show that there exists a positive number a such that if L is a linear function on Rr to R satisfying
Expert Solution
steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Fundamentals of Algebraic Equations
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax