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In Achlioptas processes, starting from an empty graph, in each step twopotential edges are chosen uniformly at random, and using some rule one of themis selected and added to the evolving graph. The evolution of the rescaled sizeof the largest component in such variations of the Erd\H{o}s--R\'enyi randomgraph process has recently received considerable attention, in particular forfor Bollob\'as's `product rule'. In this paper we establish the followingresult for rules such as the product rule: the limit of the rescaled size ofthe `giant' component exists and is continuous provided that a certain systemof differential equations has a unique solution. In fact, our result applies toa very large class of Achlioptas-like processes.  Our proof relies on a general idea which relates the evolution of stochasticprocesses to an associated system of differential equations. Provided that thelatter has a unique solution, our approach shows that certain discretequantities converge (after appropriate rescaling) to this solution.

Convergence of Achlioptas processes via differential equations with unique solutions