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Question

Let f:RR be such that f(2x1)=f(x) for all xR. 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 points
Replacing 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+...+2n12n)
f(x)=f(x2n+2n12n)
f(x)=f(x2n+112n)
Taking limit n, we get
f(x)=f(1)=1, xR.
f(x) is continuous at x=1.
f(x) is a constant function.

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