Question 5 Prove that a subset D of R is disconnected iff there exists a continuous function f: D R such that f (D) = {0, 1}.

Q: Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: Question (2): Let f,g: R R = R- 0 be any continuous functions. are they homotopic?

A: Given: Let f,g:ℝ→ℝ*=ℝ-0 be any continuous function. To check: whether f and g are homotopic or not.…

Q: Theorem 7. Let X be a continuous RV, g a function from R to R. Let Y = g(X). Then E[Y] = E[g(X)] = /…

A: The expected value of random variable is the weighted average of all possible values of the…

Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ¥x, yER. Show…

A: Introduction: Continuity is a property of a real-valued function. When the limiting value of a…

Q: 4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E…

A: given a continuous function f:ℝ→ℝ is T-periodic, defined as fx+T=fx ∀x∈ℝ to prove- for every natural…

Q: Suppose a and b are continuous functions on [a, b] and that g(x) ≥ 0 for all x ∈ [a, b]. Prove that…

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Q: Theorem 6.2.3 Let X,→X and g be a continuous function on P le bre g(X„) → g(X)

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Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

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Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

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Q: 3. Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove…

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Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

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

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Q: QUESTION 8 If f(x) is continuous on the interval [a, b], then there exists at least one number c in…

A: Here given that f(x) is continuous on a given period then need to find C

Q: (2) Let m(E) 1}) = 0. (ii) Show that lim, S f" = m({x € E : f(x) = 1}). E

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Q: (1) Suppose f: [a,b] → [a,b] is a continuous function. Prove that there exists a cE [a, b] such that…

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Q: 1 If the function f is continuous on [a,b], then there exists at least one number C in [a,b] such…

A: If the function f is continuous on [a, b], then there exists at least one number C in [a, b] such…

Q: Theorem 7.21. IfX is normal and f : X → Y is continuous, surjective, and closed, then Y is normal.

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Q: In Rolle's theorem ,If f(x) = tanx be a real valued function in [0, r],then theorem %3D A. exist B.…

A: To Determine: in Rolle's theorem,if f(x) be a real valued function in interval [0,π] Given: we have…

Q: Suppose f and g are continuous functions on [a,b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: Theorem 7.5. A function f : Rstd Rstd is continuous if and only if for every point x in R and ɛ > 0,…

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Q: 6. Let f: A R be a continuous function. Show that if A is compact then sup f(A)e f(A).

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Q: Question 4 Which among the choices is not true about the continuity of f(x)? if x 1 -{₂ f(x) = x²,…

A: I am attaching image so that you understand each and every step.

Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)=f(x) f(y) ¥x, yER. Show…

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Q: Question 4. Let f:X→Y be a mapping from a topological space (X,rx) a topological space (Y,ry). Prove…

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Q: Theorem 7.11 (Pasting lemma). Let X = A U B, where A, B are open in X. Let f : A → Y and g : B → Y…

A: Proof: Let x∈A∩B. This implies f∪g(x)=f(x)∈Yf∪g(x)=g(x)∈Y   Using the definition of agreement o…

Q: Question 1(a). Give at least three examples of continuous functions classes with justification and…

A: (a) Take f(x)=x2 in the interval [0,∞) The given function is continuous but not uniformly…

Q: f (x)dx = f(c)(b – a). a

A: given: the function f is continuous on the closed interval [a, b], then there exists at least one…

Q: Question 4. Let (fn) be a sequence of Lebesgue measurable functions on [a, b] such that fn f…

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Q: Suppose that f:[0, 1] → R is continuous on [0, 1] and differentiable on (0, 1) with f (0) = 0 and…

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

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Q: Theorem 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

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Q: Question 4. Let f:XY be a mapping from a topological space (X,rx) to a topological space (Y,ry).…

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Q: A function f is defined on [0, 1] as follows : 1 f(x)=, when 1 <x< a <*< =1, 2, 3, ..) (r=. f(0)=0,…

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Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

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Q: Find all values of a such that f is continuous on R: |x + 1 if x a

A: For a continuous function LHL ( left hand limt) should be equal to RHL (right hand limit)

Q: Theorem 12. Let f be a non-zero continuous function on R uch that f(x+y)=f(x) f(y) ¥x, yER. Show…

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Q: 10. If f(x) is continuous on [a,b], then there exists a c such that: f'(c) = ( f(b) - f(a) ) / (b -…

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Q: If f is a continuous function on [a, b], and if f (a) = -2 and ƒ (b) = 2, then there exists a real…

A: Hi, you can find the answer in the handwritten form.  Given f(a)=-2  f(b)= 2

Q: Prove that if f is a continuous function with domain [0, 1] and X is the set of all x in [0, 1] such…

A: F is a continuous function with domain (0,1) and X is the set of all x in (0,1) such that F(x) = 0.…

Q: Question 8 Let f: Z+xZ+ → Z+, where f(m,n) =m+n Is f one-to-one? Is f onto? Justify your answers.

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Q: 3. Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove…

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Q: -B- Let f( M,(Z).+.)-(M,(Z),+..) be a function define by such that g( v: al e M,(Z) . Is f is…

A: The given question is epimorphism. Given that,      f : M2ℤ , + , · → M2ℤ , + , · is a function…

Q: Suppose f: (a, b) → (a, b| is a continuous function. Prove that there exists ace Ja, b) such that C.

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Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

Q: n≥ 0 there is a unique holomorphic function h: U → C such that

A: Given, f is holomorphic in an open set U⊆ℂanda∈U.  To show that for any integer n≥0 there is a…

Q: Let f and g be continuous functions from R to R. Define the set O= {x €R: f(x)#g(x)}. If x EO, show…

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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: Prove that if f is a continuous function with domain 0, 1] and X is the set of all x in 0, 1] such…

A: Given that f is a continuous function with domain 0,1. Let, X=x| x∈0,1 and fx=0. Also, given that s…

Q: Theorem 4.12: If f: X → Y and g: Y → Z are continuous functions on the indicated spaces, then the…

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.