(7) A subset A of the real numbers is said to be dense if every open interval of the form (a,b) (where a < b) contains at least one point of A. For example, the rational numbers are a dense subset of the real numbers (so are the irrational numbers). Let A be a dense subset of R. (a) Prove that if f is continuous and f(x) = 0 for all x € A, then f(x) = 0 for all x € R. (b) Prove that if f and g are continuous functions and f(x) = g(x) for all x € A, then f(x) = g(x) for all x € R. Hint. Ilse nart (a

Q: 3. Prove if the function f : R → R is defined by the rule f (x) = 4.x + 1 for all real numbers x,…

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Q: (a) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª = 5 3 (b) Using (a),…

A: To prove  a)  If a≡b4  then prove that 3a≡3b5. b) Explain why the below function is well define…

Q: Show that [x+y] ≥ [x] + [y] for all real numbers x and y.

A: By definition, [x] ≤ x and if k be any integer such that k ≤ x, then                     k ≤ [x]

Q: Let a,b and x be real numbers with a0. Show that there exists a rational number t such that a <

A: We know that the real numbers are densely ordered. This means that between any two distinct real…

Q: 1. Let X be the set of real numbers with the usual pseudometric. Find the derived set and the…

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Q: If x and y are real numbers and x>1, prove that there exists a positive integer n such that xn > y

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Q: Determine the values of b and c such that the following function is continuous on the entire real…

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Q: Let Z be the set of integers, determine f:Z->Z, such that, for all integers a and b,…

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Q: Let f(x) = Ax² + Bx + C where A, B and C are real numbers. Prove that if f(x) is an integer whenever…

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Q: (a) Let [a, b], a <b, be any closed interval. Prove that |[a, b]| = |[0, 1].

A: Let  be any closed interval.The main aim is to prove that :"As per the policy we are allowed to…

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Q: g: Q→Q defined as g(x) = 7x+16

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Q: 3. Solve the initial value problem y" – 3y' + 2y = 10 sin x with y(0) = 1, y'(0) = -6 by using both…

A: The given initial value problem is                                          y''-3y'+2y=10sinx,…

Q: Prove if this is true of false. We know |x| = x if x ≥ 0, and |x| = -x, if x < 0. Then, for all…

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Q: Prove that if |x-1| < 1, then x² - 2x + 5| < 8.

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Q: 4. Find all real values of x such that |x-1| > |x+1]. * (-00,-1] (-00,0) [-1,0) [-1,1) None of the…

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Q: Find functions f and g from the set of positive integers to the set of real numbers such that f (n)…

A: Here, we need to find functions f and g from the set of positive integers to the set of real numbers…

Q: prove for every positive real number x that 1+1/x2>=1/x+1/x3.

A: To prove: 1+1x2≥1x+1x3; for everypositive real number x.

Q: A function f:R →→R, where R is the set of real- ax² + 5x - 8 numbers, is defined by f(x) = Find a +…

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Q: 19 a What is the domain of g (x) = sin-x + sin- V1 - x2? 1 b Show that g'(x) = Vī 2. | c Hence…

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Q: Find all positive rational numbers  x and y for which  (x+1/y)and(y+1/x)  are natural numbers.

A: x+1y,      y+1x =xy+1y and xy+1x for xy+1y &amp; xy+1x being Natural numbers x&amp; y must be 1…