$f\colon \mathbb R \rightarrow \mathbb R$ is a continuous function and $f(x)=\int_0^xf(y)~dy.$

Question

I faced the problem that says:

If $f\colon \mathbb R \rightarrow \mathbb R$ is a continuous function and $f(x)=\int_0^xf(y)~dy.$ Then which of the following option is correct?
$1.f(x)=e^x$
$2.f(x)=\ln(x)$
$3.f$ is identically $0$
$4.f$ is identically $1$.

My attempt: Here,$f'(x)=f(x)$ and so finally we can come $f(x)=Ae^x, A$ being a constant.So,option $1$ looks right.Am I right? Thanks in advance for your time.

Answer 1

Since $f$ is the Riemann integral of the continuous function $f$, it follows that $f$ is differentiable and has derivative $f$. So, as you say, $f'(x)=f(x)$, so $f(x)=A e^x$ for some constant $A$. However we also have $f(0)=\int_0^0 f(y) dy=0$. Therefore $A$ must be $0$, so $f$ is identically $0$.

Answer 2

If $f(x)$ is identically zero then also all conditions of the problem are satisfied which makes it to be one of the solution.