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A fixed-rate block code is said to be strongly robust for a class of sources if its maximum distortion over h is no larger than the maximum of the distortion-rate functions of the sources in the class at the rate of the code. It is shown that such codes exist at all positive rates whenever the class is compact with resect to the topology of weak convergence and satisfies certain additional, but not very strong, constraints on the alphabet and distortion measure. Examples of classes that satisfy these conditions are given. In addition, classes are exhibited for which there are no strongly robust codes. These help to demarcate the boundary between classes for which strongly robust codes do and do not exist. More specifically, they show that strongly robust codes exist more widely than strongly universal codes but !ess widely than weakly universal codes.