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Say you have a (continuous) map from a space A into a space Y, where A is dense in some other space X. When is it possible to extend this to a map from X into Y? Intuitively, to find the value of the map at a limit point, just take the limit of the values on a sequence approaching it. However, it's not clear the resulting sequence will always have a limit (eg, 1/x or sin(1/x) on (0,1), which is dense in [0,1)), and even when it does, is the resulting function always continuous?