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Question

If f(x) be a continuous function (a function whose graph has no breaks) defined for 1x3. 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
1
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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 10
Consider two arbitrary points x1,x2 ϵ[1,3]. Let f(x1)=q1 and f(x2)=q2, (q1,q2, ϵ Q)
Suppose q1q2. 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 q1q2 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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