(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

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

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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…

Q: g: Q→Q defined as g(x) = 7x+16

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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.

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