Show that if f and g are continuous on [a, b] and if f(a) g (b), then there exists some ce ]a, b[ satisfying f() =g (c).

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

Q: Show f e Lla, b] if and only if |f| E L[a,b].

A: The infinity norm of a function f on a,b is defined as: f∞=infk:fx≤k, almost every where on a,b.…

Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: Suppose that f : A → B is a function, C C B and D C B. Prove that Š (C – D) = } (C) - } (D).

A: Solution :-

Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: Prove that there is an additive function T : R --> R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: prove that any Continuouse function [a,b] en is measurable.

A:

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

A:

Q: Show that for any function f : X → Y , there are two functions g : S → T and h : X → S such that (a)…

A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: Count the number X of functions y: (20, 21,..., 23) → (5,6,...,8) so that is not onto. X =

A: Given function φ:(20,21,...,23)→(5,6,...,8)

Q: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: Let f and g be continuous functions on [a, b], and suppose that f(a) g(b). Prove that f(c) = alc)…

A:

Q: Let (a) Show that h is a continuous function defined on all of R.

A: Given: hx=∑n=1∞1x2+n2. We have to show that h is a continuous function defined on all of R.

Q: f(c) = g(c).

A:

Q: Let f : R → R and ECR. Prove f is continuous iff f-¹(E) is closed whenever ECR is closed.

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Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

A:

Q: Suppose f: R → R is a function. True or false: The composite function f of is continuous if and only…

A: True. Statement: Composition of f and f is continuous if and only if f is continuous. First, assume…

Q: 3. Let f A B be a function, and let Ao C A, BoC B. Prove that (a) f(f(f(Ao))) = f(Ao); (b)…

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Q: Show that the function f : R → R given by f(x) = x3 is injective.

A: To show the given function is injective(one-one), first calculate the value of function at two…

Q: If f,g continuous on [a,b] and f(a) g(b). Show that there exists c ∈ (a,b) such that f(c) = g(c)

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Q: Let f: X → Y. Prove that f is one to one if B = f -1(f(B)) for all B ⊆ X.

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Q: Is the set X of all functions f: Z+→ {0, 1} countable? Prove or disprove.

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Q: (f) Letf : A →→ B and g : B → C be functions. Prove that if g of is one-to-one, then f is…

A:

Q: Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if…

A: 1. A collection of well-defined objects or elements is called a set. If a is an element of a set A,…

Q: Let f(x,y)=g(x)h(y) where g and h are continuous functions on all real numbers such that g(x)=h(x+1)…

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Q: Prove that the function f : R → R given by f(x) = 3x° – 2x is continuous at 1 using the a.…

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Q: Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined byf(x) = ax + b, for some…

A: