4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove that f is constant.

Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

A:

Q: 31. Let f(x) = [x²/3]. Find f(S) if a) S= (-2,-1, 0, 1, 2, 3). b) S = {0, 1, 2, 3, 4, 5). c) S = {1,…

A: The given function is, fx=⌊x2/3⌋. To Find: f(S) for the following set S, (a) S=-2,-1,0,1,2,3 (b)…

Q: Let f(z) be an entire function satisfying that f(z)| ≤|z² for all z. Show that f(z) = az² for some…

A:

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: Find the values of a and b that make f continuous everywhere. b = f(x): x-2 ax² - 6x + 3 4x -a + b…

A:

Q: 1. Describe all functions f such that f(r) ar² + bx +c and f(1) = 2 and f(-1) = 6.

A:

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: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

A: The given problem is to prove the following statement. Given statement is |xy|=|x||y| for all real…

Q: Z is the set of integers, R is the set of real numbers. 3) Each function f is defined as f: R -> R.…

A: The objective is to find the function is one-one or not one-one. (a) fx=x+1x-1for all real numbers x…

Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: b) Is f one-to-one? Answer yes or no and explain. Do not just quote the definition, but give a…

A:

Q: 호호 -^ z-vz44 Z

A:

Q: Select the following functions that are bijective. ƒ₂: R → R, ƒ₂(x) = 14 f₂: R → R, f₂(x) = x f₁: R…

A:

Q: c) Is f onto? Answer yes or no and explain. Do not just quote the definition, but give a reason that…

A:

Q: 4. Show that the function f: R → R defined by f(x) = x² - 3x + 1 is not one-to-one.

A:

Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

A:

Q: a) Find the image set f (S) for S = {-2, 5}.

A:

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

A: