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Question

# If f(x) be a continuous function (a function whose graph has no breaks) defined for 1≤x≤3. f(x) ϵ Q ∀ x ϵ [1,3] and f(2)=10 (Where Q is a set of all rational numbers). Then, f(1.8) is

A
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B
5
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C
10
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D
20
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Solution

## The correct option is C 10Consider two arbitrary points x1,x2 ϵ[1,3]. Let f(x1)=q1 and f(x2)=q2, (q1,q2, ϵ Q) Suppose q1≠q2. Since f(x) is continuous, f(x) must take all the values between q1 and q2. There are infinite irrational numbers between any two rational numbers ∴ f(x) must take irrational values But f(x) ϵ Q ∀ x ϵ [1,3] This is a contradiction. Hence our assumption q1≠q2 is false. ∴ q1=q2 Since q1 and q2 are arbitrary, f(x) is a constant for all x. ∵ f(2) =10, f(x) =10 ∀ x ϵ[1, 3] ∴ f(1.8) = 10.

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