Sure, two partial functions are equivalent over a set d iff they halt on the same subset of inputs and have equal outputs whenever each terminates. Then, if you

Sure, two partial functions are equivalent over a set d iff they halt on the same subset of inputs and have equal outputs whenever each terminates. Then, if you could decide equality of two functions, f, g, over a finite domain d, then let g be a function that loops infinitely for any input, and check equality of f and g over a singleton containing some input. If f doesn't halt, then they're equal (since they both don't halt over the same set) and if f halts then they're not equal since the sets they halt on don't agree.