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(C) Given the equation $f(x) = x + \int_0^x f(t) \, dt$.By applying the Fundamental Theorem of Calculus and differentiating both sides with respect to

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$f(x+y) = f(x) + f(y) + f(x)f(y)$

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(C) Given the equation $f(x) = x + \int_0^x f(t) \, dt$.By applying the Fundamental Theorem of Calculus and differentiating both sides with respect to $x$,we get $f'(x) = 1 + f(x)$.This is a linear differential equation: $f'(x) - f(x) = 1$.The integrating factor is $e^{\int -1 \, dx} = e^{-x}$.Multiplying by the integrating factor: $e^{-x} f'(x) - e^{-x} f(x) = e^{-x}$.Integrating both sides: $\frac{d}{dx} (e^{-x} f(x)) = e^{-x}$.$e^{-x} f(x) = \int e^{-x} \, dx = -e^{-x} + C$.$f(x) = -1 + C e^x$.Since $f(0) = 0 + \int_0^0 f(t) \, dt = 0$,we have $0 = -1 + C$,so $C = 1$.Thus,$f(x) = e^x - 1$.Now,consider $f(x) + f(y) + f(x)f(y) = (e^x - 1) + (e^y - 1) + (e^x - 1)(e^y - 1)$.$= e^x - 1 + e^y - 1 + e^{x+y} - e^x - e^y + 1$.$= e^{x+y} - 1 = f(x+y)$.Therefore,$f(x+y) = f(x) + f(y) + f(x)f(y)$.

$A$ continuous function $f: R \rightarrow R$ satisfies the equation $f(x) = x + \int_0^x f(t) \, dt$. Which of the following options is true?

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