3.1 Let S be the set of all bounded continuous functions f: [a, b] → R. Given two points f and g in X, let d(f. g) = sup{f(x) - g(x) : x = [a, b]} Determine whether d is a metric on S or not.

Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: Which of these functions from C[0, 1] to R (with the usual metric) are continuous?

A: Given: Functions are defined from C0, 1 to R. To find whether the following functions are…

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Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

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Q: 3. Let f, g : [a, b] → [c, d] be cotinuous functions. Suppose that f is surjective. Prove that there…

A: We will use of intermediate value theorem to proof this

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A: We will use the fact that the MRS of a homothetic function is always homogeneous of degree 0. Let…

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A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: Identify an inner function (call it g) of y = (5x + 4)3.Let u = g(x) and express the outer function…

A: y=(5x+4)3

Q: Show, using the definition of continuity from the topological point of view, that the constant…

A: See the attachment.

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given : As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a…

Q: . If f : (0, 3) → R is continuous, must its range be an open (possibly infinite) interval?

A: See the detailed solution below.

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Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

Q: 4. Let X and Y be metric spaces and f : X → Y be continuous. Let SC X. Show: (b) If S is compact,…

A: First, we prove that fS¯⊆f(S)¯. Given that f is continuous and S¯ is compact, we can say f-1Y-f(S)¯…

Q: Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x…

A: Given:d1:X×X→R such that  d1x, y=0if x=y2if x≠y∀x , y∈X To Show: d and d1 are equivalent.

Q: Prove, using the definition of continuity from the topological point of view, that the constant…

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Q: Let I be an interval and f : I → R be a function with Darboux property such that f(I) is finite.…

A: Let I be an interval and f : I → R be a function with Darboux propertysuch that f(I) is finite.…

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A: Matric Space

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Q: . Let X be a nonempty set with metric d. Show that the function e defined by e(x, y) = min[1, d(x,…

A: We know that d(x,y) is a metric on X. Hence d satisfies all the properties of a metric.  Now e(x,y)…

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Q: Determine the domain and range of the function f: {r,s,t,u} ➔ {A,B,C,D,E} defined by f(r) = A, f(s)…

A: Domain of a function is the set of possible values of independent variable. The domain of given…

Q: Let f: R R be the absolute value function and let g : R -→ R be an affine map for which g(0) = 0.…

A: Given f:ℝ→ℝ is the absolute value function, that is,                  fx=x, for all x∈ℝ and g:ℝ→ℝ is…