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The question is what set of continuous functions solves the problem $g(x)=\int_{3}^{x} g(t) dt$. My answer so far: g(3)=0, g'(x)=g(x)-g(3) therefore g(x)=g'(x)=$ce^x$. Obviously $ce^x=ce^x-ce^3$ doesn't really work out. Where's my mistake? And what's the correct answer? Thanks.

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Take the derivative we find $$g'(x)=g(x)$$ hence $$g(x)=\lambda e^x$$ and since $g(3)=0$ then $\lambda=0$ hence $g=0$.