Suppose f : X → Y is a function. Prove that f is surjective if and only if for all sets Z and all functions g, h : Y → Z, go f = ho f implies g = h.

Q: f,(x)= e* f2(x)=e¯* (x) = e'

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Q: Let h: R → R be a continuous function on all of R. If h(x) Tt, prove that h(t) 50.

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Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

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Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

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Q: Prove that the function f : R → R, given by f(x) = 2x, is one-one and onto.

A: Given, function f : R → R, given by f(x) = 2x.

Q: Define f : R → R by f(x) = 0 if x∈ R\Q f(x)=1/q if x=p/q with (p, q) = 1. Prove that f is…

A: Define f:ℝ→ℝ byf(x)=0if x∈ℝ\ℚ1qif x=pq with (p,q)=1Prove that f is continuous on ℝ\ℚ and…

Q: Suppose f : [a, b] → [a, b] is continuous. Prove there exists c E [a, b] such that f(c) = c.

A: Now given f : a,b→a,b continuous gx=fx-x⇒ gx is continuous By Algebra of continuousga=fa-a ≥0gb=fb-b…

Q: Let f : (0, ∞) → R be defined by f(x) = ln x. Prove that f is onto.

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Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

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Q: Define f : Zmn → Zm × Zn by ƒ ([x]mn) = ([x]m, [x]n). Show that f is a function and that f is onto…

A: The function f:ℤmn→ℤm×ℤn is defined by                  fxmn=xm,xn Show f is a function and f is…

Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

Q: If f is continuous on [a, b], then there is cE[a, b] such that f(c) = av(f). Select one: True False

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Q: Let X,Y,Z be sets and f:X→Y, g:Y→Z. Prove that if g is injective and g◦f is surjective,then f is…

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Q: Letf:E→R be uniformly continuous. (a) Prove that E is bounded⇒f(E) is bounded.

A: Given, f:E→ℝ is uniformly continuous. Since, E is a bounded subset of ℝ and f is uniformly…

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(Tt) <0.

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Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

Q: Prove the following statement: For any function f : R" - R and any function g : R" R, let h: R" R be…

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Q: Let h: R → R be a continuous function on all of R. If h(x) I prove that h(Tt) <0.

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Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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Q: Suppose that f is a continuous function satisfying stæ) = a f(t)dt + * So f(t)dt + a %3D for all a,…

A: Solve the function

Q: : Iff (z) is analytic at zo. (a) prove that it must be continuous at zo.

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Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. O True O False

A: The given function is                                     f: ℤ×ℤ→ℤ  such that    f(m,n)=m-n. 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: For a bijective function f:(X,r1)→ (Y,12), one of the following statements is true: O f is open if…

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Q: Prove that f (z) =zz is nowhere analytic

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Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. True False

A: Here we will examine that f is onto or not.

Q: Let P denote the set of functions f in C([0,1]) with f(x) > 0 for all x. Then P is a closed subset…

A: Given that, P denotes the set of functions f in C0,1

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

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Q: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

A: Definition: An open map is a function between two topological spaces that maps open sets to open…

Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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Q: Let f:A→B. Prove that if X⊆A and f is injective, then f(A−X) =f(A)−f(X)

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Q: Consider a function f:C C, defined by f(z) = 2z2 Then which of the following is true? Check all that…

A: Injective ( one to one): If x1 not equal to x2 then fx1 not equal to fx2. If x1 = x2 then fx1 = fx2.…

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

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Q: Let f : R → R be continuous such that f(q) = 0 for all q ∈ Q. Prove that f must be the zero function

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Q: Let f, g : D → R be continuous at c E D. Prove that fg is continuous at c.

A: Let ε&lt;1 and M= maxf(a), g(a), 1, then there exists δ&gt;0 such that f(x)-f(a)&lt;ε3M (since f is…

Q: If f(x) and g(x)are continuous on [a,b]and if [f = [8 . Prove that 3ce[a,b] such that f(c)= g(c).

A: Here, we will going to use Intermediate Value  Theorem to prove this result

Q: Define f: R → [2, 0) by f(x) = x² +2,Vx € R.Prove that f is onto.

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Q: Suppose f(z) is analytic for |z| &lt;= 1, and |f(z)| &gt;=1. If f(0) =1, prove that f(z) is…

A: To prove that the given conditions on the analytic function force it to be the constant function…

Q: Complex analysis

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Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

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Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

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Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: Suppose f is differentiable on (a, b). If c E (a, b) and f' (c) = 0, thenf (c) is either the maximum…

A: Given f is differentiable on (a,b). and c lies in (a,b) such that f'(c)=0.

Q: Consider the function h : Z x Z → Z ×Z defined by h(x, у) — (2ху, 7x — Зу). (a) Is h injective? (b)…

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Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊m/n ⌋

A: Given that: f(m,n)=mn,  f:Zx(Z-{0})→Z

Q: Let f:C → C be an entire function. Show that the function g:C \ {0} → C defined by g(z) = f(1/7) is…

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Q: (c) Suppose in addition that the set D = {r €R: g(r) = 0} is finite and ro ¢ D. Define a function e…

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