Homework Answers
A function on R does not have extreme values (that is neither maximum nor minimum ) by three ways :
1.if it is constant
2.it is always increasing
3.it is always decreasing
Now we take an easy example , the contstant function, f(x) =C. Which is continuous and bounded on R but doesn't have minimum or maximum values (because its derivative is zero).
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an example of a continuous and bounded function on all of R that does not attain...
Construct a continuous function f : D -> R which is bounded and does not attain...
Construct a continuous function f : D -> R which is bounded and does not attain 6. its maximum, if D = Bi(0) C R2. Can one construct such a function in the case of the close ball D B1(0)?
Construct a continuous function f : D -> R which is bounded and does not attain 6. its maximum, if D = Bi(0) C R2. Can one construct such a function in the case of the close ball D B1(0)?
Question 2.1. . (i) Give an example of a function, f: R R, that is not...
Question 2.1. . (i) Give an example of a function, f: R R, that is not bounded. (ii) Give an example of a function, f: (1.2) + R, that is not bounded. (iii) Give an example of a function, f: R → R. and a set. S. so that f attains its maximum on S. (iv) Give an example of a function, f: R R , and a set, S, so that f does not attain its maximum on S....
11. Consider the function f(x) = e-l=\. Where does f attain its absolute maximum on all...
11. Consider the function f(x) = e-l=\. Where does f attain its absolute maximum on all of C? Why does this not contradict the maximum modulus principle?
Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that t...
Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that the limits limoo f(x) and lim f(x), are well-defined. (ii) Give a counterexample of a bounded, continuous, compactly supported function f that is not of bounded variation.
Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that the limits limoo f(x) and lim f(x), are well-defined. (ii) Give a counterexample of a bounded, continuous, compactly supported...
Find a continuous function f: (0,1] → R that does not have a minimum.
Find a continuous function f: (0,1] → R that does not have a minimum.
3. Let the function f be a real valued bounded continuous function on R. Prove that...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
B) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) su...
b) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) such that R If(x) dz 00. Give an example of a sequence {fn} of functions in D(0,00)) which (i) converges pointwise for E [0, oo) to the constant function f(z)0 (ii) does not converge to 0, neither with respect to the norm, nor the Hint: it may be helpful to contemplate the phrase "mass escaping to infinity". norm.
b) (10 pts) Let...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a,...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
(1)Give an example of a function f : (0, 1) → R which is continuous, but...
(1)Give an example of a function f : (0, 1) → R which is continuous, but such that there is no continuous function g : [0, 1] → R which agrees with f on (0, 1). (2)Suppose f : A (⊂ Rn) → R. Prove that if f is uniformly continuous then there is a unique continuous function g : B → R which agrees with f on A.(B is closure of A)
I need to answer 1b 2.5. Let f be a real valued function continuous on a...
I need to answer 1b
2.5. Let f be a real valued function continuous on a closed, bounded Theorem set S. Then there exist x1,X2 S such that f(x1) S f(x) s f(x2) for all x e S. Proor. We recall that if T E' is bounded and closed, then y, - inf T and sup T are points of T (Example 4, Section 1.4). Let T- fIS. By Theorem 2.4, T is closed and bounded. Take x, such that...
Construct a continuous function f : D -> R which is bounded and does not attain 6. its maximum, if D = Bi(0) C R2. Can one construct such a function in the case of the close ball D B1(0)?
Construct a continuous function f : D -> R which is bounded and does not attain 6. its maximum, if D = Bi(0) C R2. Can one construct such a function in the case of the close ball D B1(0)?
Question 2.1. . (i) Give an example of a function, f: R R, that is not bounded. (ii) Give an example of a function, f: (1.2) + R, that is not bounded. (iii) Give an example of a function, f: R → R. and a set. S. so that f attains its maximum on S. (iv) Give an example of a function, f: R R , and a set, S, so that f does not attain its maximum on S....
11. Consider the function f(x) = e-l=\. Where does f attain its absolute maximum on all of C? Why does this not contradict the maximum modulus principle?
Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that the limits limoo f(x) and lim f(x), are well-defined. (ii) Give a counterexample of a bounded, continuous, compactly supported function f that is not of bounded variation.
Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that the limits limoo f(x) and lim f(x), are well-defined. (ii) Give a counterexample of a bounded, continuous, compactly supported...
Find a continuous function f: (0,1] → R that does not have a minimum.
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
b) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) such that R If(x) dz 00. Give an example of a sequence {fn} of functions in D(0,00)) which (i) converges pointwise for E [0, oo) to the constant function f(z)0 (ii) does not converge to 0, neither with respect to the norm, nor the Hint: it may be helpful to contemplate the phrase "mass escaping to infinity". norm.
b) (10 pts) Let...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
I need to answer 1b
2.5. Let f be a real valued function continuous on a closed, bounded Theorem set S. Then there exist x1,X2 S such that f(x1) S f(x) s f(x2) for all x e S. Proor. We recall that if T E' is bounded and closed, then y, - inf T and sup T are points of T (Example 4, Section 1.4). Let T- fIS. By Theorem 2.4, T is closed and bounded. Take x, such that...
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