Let \( f, g : \mathbb{R} \to \mathbb{R} \) be continuous at a point \( c \), and let \( h(x) := \sup \{ f(x), g(x) \} \) for \( x \in \mathbb{R} \). Show that \[ h(x) = \frac{1}{2}(f(x) + g(x)) + \frac{1}{2}|f(x) - g(x)| \]for all \( x \in \mathbb{R} \). Use this to show that \( h \) is continuous at \( c \).

Answered: Image /qna-images/answer/95dd242b-e57d-4b13-a8d5-1434f6744fe4.jpg

Answered: Let f, g: R → R be continuous at a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) = f(|x| + 2). A student presents the following proof to show that there exists a real number c € (-1, 1) such that g/(c) = 0. (1) Since f is a continuous function, so is g over the interval [-1, 1]. (II) Since f is differentiable, so is g over the interval (-1, 1). (III) It is evident from the definition of g that g(−1) = g(1). (IV) If the above conditions hold, then by Rolle's theorem, there exists dg(x) c = (-1, 1) such that gl (c) dx |x=c Which statement about this proof is correct? = 0. Step (1) does not hold, and hence Rolle's theorem does not apply. Step (II) does not hold, and hence Rolle's theorem does not apply. Step (III) does not hold, and hence Rolle's theorem does not apply. Step (IV) does not hold, and hence the conclusion is false. The proof is completely correct, and the conclusion holds.
Use the three-steps definition of continuity to show that the function f(z) = z² - 2z+ (1 + i) is continuous at zo = 3 - i.
1. Explain what it intuitively means for function f(x, y) to be continuous in some domain D.
Use FTC I to prove that if |f '(x)| ≤ K for x ∈ [a, b], then |f (x) − f (a)| ≤ K|x − a| for x ∈ [a, b].
Show that the function f : R → R given by f(x) = x/(1+|x|) is increasing.
Use the e-d definition to prove that f(x) = x³ + 3 is continuous at xo = 3.
Prove that the function f: R → R given by f(x) = 3x³ – 2x is continuous at 1 using the a. sequential definition, b. e - 8 definition.
Let f and g be continuous functions from R to R. Show that the function F(x) = min{f(x), g(x)} is a continuous function.

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,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

Let f, g: R → R be continuous at a point c, and let h(x) = sup{f(x), g(x)} for x € R. Show that h(x) = (f(x) + g(x)) + ½ |ƒf(x) − g(x) for all x € R. Use this to show that h is continuous at c.