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Question

Let f:RR be a function such that f(x+y2)=f(x)+f(y)2 for all x, y, and f(0)=3 and f(0)=3. Then

A
f(x)/x is continuous on R
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B
f(x) is continuous on R
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C
f(x) is bounded on R
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D
none of these
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Solution

The correct option is A f(x) is continuous on R
limh0f(x+h)=limh0f(2x+2h2)

=limh012[f(2x)+f(2h)]

=12f(2x)+12limh0f(2h)=12f(2x)+12f(0)

(f is differentiable at 0 so continuous also)
Putting y=0 in the given equation, we have
f(x)=f(2x2)=f(2x)+f(0)2

Hence limh0f(x+h)=f(x)

Since f(x)/x is not defined at x=0,f(x)/x is not continuous on R. Clearly f(x) need not be bounded on R.

e.g. f(x)=x satisfies the given equation but is not bounded on R.

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