I once read a Microsoft research paper. I think its central premise had a fatal flaw:

http://research.microsoft.com/en-us/um/people/lamport/pubs/b...

The author confused continuity of the DECISION FUNCTION with continuity of the OUTCOME CURVE.

In other words, an algorithm such as "keep trying to make a decision until you notice that t > C, after which always pick the left one" will in fact NOT have a continuous outcome curve, despite the decision function being continuous.