Suppose that a continuous real-valued function \(f(x)\) of real numbers satisfies \(f(x + xy) = f(x) + f(xy)\) for all real \(x, y,\) and that \(f(2020) = 1\). Compute \(f(2021)\).
Question and options
Suppose that a continuous real-valued function of real numbers satisfies for all real and that . Compute .
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Answer and explanation
Correct answer: A - \(\frac{2021}{2020}\)
Put u=x, v=xy (any u not 0, any v): f(u+v)=f(u)+f(v); x=0 gives f(0)=0. A continuous additive function is linear, f(x)=kx. f(2020)=1 gives k=1/2020, so f(2021)=2021/2020.
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