To see why A1-A4 is not enough to prove C4 on its own, consider the preference relation on the space of lotteries between two outcomes X and Y such that all lotteries are equivalent if P(X)≤12, and if P(X)≥12, higher values of P(X) are preferred. This satisfies A1-A4, but cannot be expressed with a concave function, since we would have to have u(X+Y2)=u(X)<u(X)+u(Y)2, contradicting concavity. We can, however express it with a quasi-concave function: U(pX+(1−p)Y)=max(0,p−12).
To see why A1-A4 is not enough to prove C4 on its own, consider the preference relation on the space of lotteries between two outcomes X and Y such that all lotteries are equivalent if P(X)≤12, and if P(X)≥12, higher values of P(X) are preferred. This satisfies A1-A4, but cannot be expressed with a concave function, since we would have to have u(X+Y2)=u(X)<u(X)+u(Y)2, contradicting concavity. We can, however express it with a quasi-concave function: U(pX+(1−p)Y)=max(0,p−12).