In constructive mathematics, one sometimes calls a function ‘extensional’ to stress that it preserves equality (as opposed to a prefunction, which need not do so). A function is strongly extensional if it also reflects inequality in a relevant sense. One usually employs this in a context where the principle of excluded middle is enough to prove that every function is strongly extensional.
This is probably not the only context where the term ‘strongly extensional’ is used, but it's the one that I know.
Note that if is only a prefunction (in a foundation where this makes sense) and we impose the condition of strong extensionality, it follows (upon taking the contrapositive?, since equality is the negation of any tight apartness) that is extensional (and so a function). Conversely, if is also the negation of equality (which is always the unique tight apartness in classical mathematics), then any extensional is also strongly extensional. In this way, strong extensionality is a constructively stronger but classically equivalent variation of extensionality.
A strongly extensional function is strongly injective if it also preserves . (These are the regular monomorphisms of , the monomorphisms being merely injective in the usual sense of reflecting equality.)
Any pointwise-continuous function between metric spaces is strongly extensional. Using countable choice and Markov's principle (or actually weak versions thereof), it follows that any function to a metric space from a complete metric space (regardless of continuity!?) is strongly extensional.
A weak counterexample: Internal to the topos of sheaves on the unit interval , where an internal real number is (externally) a continuous map from to the real line , consider the internal number given by the function . Then the subspace of the internal real line is an internal complete metric space (as proved by Richman), we may define an internal continuous map by and , and of course we have internally. But the internal truth value is only the open interval , not all of , so it's not completely true that is strongly extensional. (However, this is not not strongly extensional, so a stronger counterexample would be nice.)