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Question

Let f:R→R be such that f(2x−1)=f(x) for all x∈R. If f is continuous at x=1 and f(1)=1, then

A
f(2)=1
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B
f(2)=2
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C
f is continuous only at x=1
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D
f is continuous at all points
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Solution

The correct options are A f(2)=1 D f is continuous at all pointsReplacing x by (x+12) f(x)=f(x+12) Again, we put (x+12) in place of x, we get f(x+12)=f⎛⎜ ⎜ ⎜⎝x+12+12⎞⎟ ⎟ ⎟⎠=f(x+1+222) Again, we put (x+12) in place of x, we get f(x+1+222)=f(x+1+2+2223) Repeating this process n times , we get f(x)=f(x+1+2+22+23+...+2n−12n) ⇒f(x)=f(x2n+2n−12n) ⇒f(x)=f(x2n+1−12n) Taking limit n→∞, we get f(x)=f(1)=1, ∀ x∈R. f(x) is continuous at x=1. ⇒f(x) is a constant function.

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