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Question

Let f(x+y)=f(x)+f(y) and f(x)=x2g(x) for all x,yϵR, where g(x) is continuous function. Then f(x) is equal to

A
g'(x)
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B
g(0)
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C
g(0) + g'(x)
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Solution

We have f(x)=limh0f(x+h)f(x)h=limh0f(x)+f(h)f(x)h
[f(x+y)=f(x)+f(y)]

=limh0f(h)h=limh0h2g(h)h=0.g(0)=0
[because g is continuous therefore limh0 g(h)=g(0)].


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