Question: Suppose that E is a compact subset of X. a) If f, g : E Rn are uniformly continuous, prove that f + g

Suppose that E is a compact subset of X.
a) If f, g : E → Rn are uniformly continuous, prove that f + g and f ∙ g are uniformly continuous. Did you need compactness for both results?
b) If g: E → R is continuous on E and g(x) ≠ 0 for x ∈ £, prove that 1/g is a bounded function.
c) If f, g: E → R are uniformly continuous on E and g(x) ≠ 0 for x ∈ E, prove that f/g is uniformly continuous on £.

Step by Step Solution

★★★★★

3.46 Rating (169 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Repeat the proof of Exercise 345ab Compactness was used to prove fg is uniformly continuous ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (602).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Suppose that is a compact subset of R. If for every x E there exist a nonnegative function f = fx and an r = r(x) > 0 such that f is C on R, f(t) = 1 for t (x - r, x + r), and f(t) = 0 for t (x -...

Suppose that (E) is a Cp surface and that (x0, y0, z0) = (u0, v0)* where (u0, v0) E. If N (u0, v0) 0, prove that (E) has a tangent plane at (x0, y0, z0)-

Suppose that E is a nonempty subset of R and that fn f uniformly on E. Prove that if each fn is uniformly continuous on E, then f is uniformly continuous on E.

7. Suppose that E is a compact subset of a metric space X. (a) If I,g : E -+ Rn are uniformly continuous, prove that 1+ 9 and I 9 are uniformly continuous. Did you need compactness for both results?...

6. Suppose that E is a nonempty subset of R and that f n -+ f uniformly on E. (a) Prove that if each fn is continuous on E, then f is continuous on E. (b) Prove that if each f n is uniformly...

. Answer the following questions. 1. (3 points) In Metropolis only taxicabs and privately owned automobiles are allowed to use the highway betwem the airport and downtown. The market for taxi cab...

Math 636 Final - Quiz Component 1. The system of linear equations 2x2 x3 = 3 2x1 + x2 + 2x3 = 1 x1 + x2 + 3x3 = 2 (a) is consistent with a unique solution. (b) is consistent with infinitely many...

Suppose that a R, that is a nonempty subset of R, and that f, g : E R are uniformly continuous on E. a) Prove that f + g and f are uniformly continuous on E. b) Suppose that f, g are bounded on E....

Let E Rn and suppose that D is dense in E (i.e., that D E and = E). If f: D Rm is uniformly continuous on D, prove that f has a continuous extension to E; that is, prove that there is a...

Suppose that X is a separable metric space which satisfies the Bolzano-Weierstrass Property, that Y is a complete metric space, and that E is a bounded subset of X. Prove that a function f: E Y is...

Find vo(t) for t > 0 in the circuit in figure and plot and response including the time interval just prior to closing the switch. 12 V 1H ( 1)900 400

Match the following integrals with the verbal descriptions of the solids whose volumes they give. Put the letter of the verbal description to the left of the corresponding integral. 1-3y2 V1 4x2 3y...

The risk that a borrower, also known as a creditor, might not _ _ _ _ _ is called credit risk or default risk. A . repay a loan B . buy insurance C . pay the specified interest rate D . send...

Individual actions can play a large role in the overall health of our planet. A researcher interested in evaluating environmentally friendly behaviors evaluated how often people recycle (per month)...

1. Find the interval of convergence of each of the following power series. 00 k (a) L~k. 00 (b) L(( _l)k + 3)k(x - l)k. k=O k=O 00 (k+1) (c) Llog -k- xk. k=l * ~ 13 (2k - 1) 2k (d) ~ (k I)! x. k=l +

*9. Suppose that ak ! 0 as k -t 00. Prove that L::;;O=l ak sin kx converges uniformly on any closed interval [a,b] C (0, 21r).

8. Let n ~ 0 be a fixed nonnegative integer and recall that O! := 1. The Bessel function of order n is the function defined by 00 (-l)k (x)n+2k Bn(x) := {; (k!)(n + k)!"2 . (a) show that Bn (x)...

1. An analyst's factor model for Stock V is rt = 0.030 + 1.90ft + et. Calculate the residual for Stock V if actual return is 0.103 and ft = 0.058. Express your answer as a decimal with four digits...

1-7 The following data was obtained for a reactant for the reaction + . Determine the order of the reaction for the reactant and the value of the reaction rate constant k.

(15 points) Suppose that projects cash flows are C0 = $82, C1 = $30, C2 = $100, C3 = $100, C4 = $150. Required rate of return is 10%. (a) (5 points) Compute NPV. Should you take the project based on...