2. Show that, on the space of continuous functions C[0, 1] the distances defined by the norms =√₁|f(t)\Pdt, ‚ = max |ƒ(x)\, ||f||1₁ = [₁ |ƒ(t)|dt, ||f||2= ||f||∞ = are mutually non-equivalent.

Q: Every bounded continuous function attains its maximum and minimum values.

A: Here’s a sketch of one possible argument. Let u=supx∈[a,b]f(x). (Note that I allow the possibility…

Q: [1+2p(x-1)+(x-1)² x≤1 x>1 [qx+p -) -) Find the values of q, in terms of p, for which fis continuous…

A: Given:f be a function defined by:a. To find the value of q in terms of p, for which f is…

Q: Suppose that a 0. Prove f() is bounded. that the function } : [a, b] → R defined by () (x) = =

A:

Q: Let f ={ (-6,4),(-5,3),(4,-6),(3,-5),(1,5),(2,6),(-3,5),(-8,2),(6,4)}  g=…

A: It is known that f-gx=f(x)-g(x) and f·gx=f(x)·g(x). The sets are provided as:…

Q: Find the absolute maximum and minimum values of f on the set D. f(x,y) = x2 + y2 + x2y +6 D =…

A:

Q: = √x + 2,-2 ≤ x ≤2 ax²-bx, x+1 x-2 ) 2 3

A: Topic:- calculus

Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

A:

Q: Consider the three basic vectors: (2,1,0,-1),(-1,1,1,1),(2,7,4,1) a. give two vectors that are in…

A: Given the three basic vectors:                           (2,1,0,-1),(-1,1,1,1),and (2,7,4,1) We have…

Q: If x + 1 f(x) then find the global minimum of f on the interval [1, c∞)

A: Given: Function fx=x2+1x on the interval [1,∞) To find: Global minimum of given function. Concept:…

Q: Let A, B C X be subsets and f : X → Y a function. Show that ƒ(AÑ B) ≤ f(A)Ñ ƒ(B)

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Q: (a) Assume f: S → R is uniformly continuous and S is bounded. Prove that the range of f is bounded.…

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Q: Let f be continuous on an interval I from a to b with only one critical number c in (a, b). If f'(c)…

A: Step 1: Step 2: Step 3: Step 4:

Q: 1. Given a set X and a subset A CX, define a function ƒÃ : X → {0, 1} by setting f₁(x) 1 if ¤ ¤ A…

A: (a) fA : X → {0, 1} is defined as             fA(x) = 1 when x ∈ A                       = 0 when x…

Q: 2. Let f be a continuous function defined on a closed, bounded interval I = [a, b]. Assume that f is…

A: Intermediate Value Theorem- Let f be a continuous function over a closed interval [a, b] with its…

Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

A:

Q: The linear functional F: R^2[x] -> R is defined by the formula F: p -> ∫(from 1 to 0) p(t)dt.…

A: Step 1: Step 2: Step 3: Step 4:

Q: Let's assume that f(1) = 1² + c* I c=14 g(z, a) = a* z²+a*z Find a such that the following error is…

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

A:

Q: 3. The function f(m, n) = 2"3" is a one-to-one function from N x N into N. (a) Calculate f(m, n) for…

A:

Q: 5. Assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove there are x, y = [0, 1] with |x -…

A: We have to assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). and prove that there are x, y ∈…

Q: Only one can be true

A: since f(a)>0, so there is at least x=a , be the point at which function value is positive. Hence,…

Q: (2) Show that C(0, 1] C L'(0,1]. Compare |flsup and fl. Justify your answer. Generalize this to the…

A:

Q: 2. Sketch the graph of one function g that is continuous on [-4, 4] and has all of the following…

A: Solved below.