# Two formulations of the continuity axiom in Expected Utility Theory

MWG state the continuity axiom as follows:

C1. $$\succsim$$ is continuous if for all $$L,L',L''$$, the two sets below are both closed: \begin{align} S&=\{\alpha\in[0,1]:\alpha L+(1-\alpha)L''\succsim L'\}\\ T&=\{\alpha\in[0,1]:L'\succsim \alpha L+(1-\alpha)L''\} \end{align}

Other authors (e.g. Kreps, Rubinstein, Levin) use a somewhat different formulation:

C2. $$\succsim$$ is continuous if for all $$L,L',L''$$ with $$L\succsim L'\succsim L''$$, there exists an $$\alpha\in[0,1]$$ such that $$$$L'\sim \alpha L+(1-\alpha)L''$$$$

Are the two formulations equivalent? It's easy to see how C1 implies C2 (just take $$\alpha\in S\cap T$$). But I'm not sure how C2 implies C1.

The proofs using C2 usually first establish that, with independence, we have $$$$\beta L+(1-\beta)L'\succ \alpha L+(1-\alpha)L'$$$$ whenever $$L\succ L'$$ and $$1>\beta>\alpha>0$$. Hence, C2 and independence together imply C1. But does C2 imply C1 without independence?