2. Define, for x E S [0,1] U [2,3), the 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![Theorem 4.4.3. Suppose that f is continuous and one-to-one on
[a,b]. Then f1 is continuous on f([a,b]).
Proof. From the previous theorem, f is either strictly increasing or
decreasing on [a, b]. Assume that f is strictly increasing. (Applying
these considerations to the function -f if f were decreasing would
complete the proof.) Take any point yo E (f(a), f(b)) and any ɛ > 0.
We need to show that there is a ō> 0 such that
ly– yol < 8
implies w)-f0)| < e.
Set xo = f'(yo) E (a,b). Replacing the given ɛ with a smaller value
if necessary, we can assume that a s xo - E < Xo + ɛ < b. Then f(xo
- E) < f(xo) < f(Xo + ɛ), and from the previous theorem, f1 is strictly
increasing on [f(Xo – €), f(xo + ɛ)], mapping this interval into [Xo – E,
Xo + ɛ]. Hence, let
|
8 = Min {yo - f(xo – €), f(xo+E) – yo}
(see Figure 4.14). If yo =
argument to the interval [a, a + ɛ] (or [b – ɛ,b]).
f(a) (or f(b)), we modify the above
Thus, we have that, for a function f one-to-one on an interval [a,
b], continuity of f implies the continuity of the inverse. We now use
this result in our discussion on differentiability.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ab050a-d13e-4a73-9fee-dc44d5d071e8%2F40984563-5fc6-481c-b276-9c7e0ae60623%2F9lcjn7_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 4.4.3. Suppose that f is continuous and one-to-one on
[a,b]. Then f1 is continuous on f([a,b]).
Proof. From the previous theorem, f is either strictly increasing or
decreasing on [a, b]. Assume that f is strictly increasing. (Applying
these considerations to the function -f if f were decreasing would
complete the proof.) Take any point yo E (f(a), f(b)) and any ɛ > 0.
We need to show that there is a ō> 0 such that
ly– yol < 8
implies w)-f0)| < e.
Set xo = f'(yo) E (a,b). Replacing the given ɛ with a smaller value
if necessary, we can assume that a s xo - E < Xo + ɛ < b. Then f(xo
- E) < f(xo) < f(Xo + ɛ), and from the previous theorem, f1 is strictly
increasing on [f(Xo – €), f(xo + ɛ)], mapping this interval into [Xo – E,
Xo + ɛ]. Hence, let
|
8 = Min {yo - f(xo – €), f(xo+E) – yo}
(see Figure 4.14). If yo =
argument to the interval [a, a + ɛ] (or [b – ɛ,b]).
f(a) (or f(b)), we modify the above
Thus, we have that, for a function f one-to-one on an interval [a,
b], continuity of f implies the continuity of the inverse. We now use
this result in our discussion on differentiability.
![2. Define, for x e S = [0,1] U [2,3), the function
%3D
0<x<1
f(x) =
14-x,
2<x<3'
(a) Show that f is continuous and one-to-one on S.
(b) Show that T = f(S) = [0,2],
(c) Determine f'(y) for y e T.
(d) Show that f1 is not continuous at 1.
(e) Why doesn't this example contradict Theorem 4.4.3?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ab050a-d13e-4a73-9fee-dc44d5d071e8%2F40984563-5fc6-481c-b276-9c7e0ae60623%2Ffnrfp2k_processed.png&w=3840&q=75)
Transcribed Image Text:2. Define, for x e S = [0,1] U [2,3), the function
%3D
0<x<1
f(x) =
14-x,
2<x<3'
(a) Show that f is continuous and one-to-one on S.
(b) Show that T = f(S) = [0,2],
(c) Determine f'(y) for y e T.
(d) Show that f1 is not continuous at 1.
(e) Why doesn't this example contradict Theorem 4.4.3?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Knowledge Booster
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
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
