Let $f$ be a function satisfying $f(x+y) = f(x) f(y)$ with $f(0) = 1$ and $g$ be a function that satisfies $f(x) + g(x) = x^2$. Then the value of the integral $\int \limits _0 ^1 f(x) g(x) \textrm d x$ is:
a. $\textrm e - \dfrac {\textrm e ^2} 2 - \dfrac 5 2$
b. $\textrm e + \dfrac {\textrm e ^2} 2 - \dfrac 3 2$
c. $\textrm e - \dfrac {\textrm e ^2} 2 - \dfrac 3 2$
d. $\textrm e + \dfrac {\textrm e ^2} 2 + \dfrac 5 2$
If $f$ is continuous, $f(x) = \exp(cx)$ for some constant $c$. Then $g(x) = x^2 - \exp(cx)$, and $\int_0^1 f(x) g(x)\; dx$ is a rather complicated function of $c$:
$$ {\frac {-{{\rm e}^{2\,c}}{c}^{2}+ \left( 2\,{c}^{2}-4\,c+4 \right) {{\rm e}^{c}}+{c}^{2}-4}{2 {c}^{3}}} $$
It turns out that this can be anything in the interval $(-\infty, 0)$. So (a) and (c) are possible answers, but (b) and (d) are not.
On the other hand, there are also solutions where $f$ is non-measurable. In that case, the answer is "undefined".