Is finding the equivalence of two functions undecidable?

Question

Is it impossible to know if two functions are equivalent? For example, a compiler writer wants to determine if two functions that the developer has written perform the same

Answer 1

In the general case it's undecidable whether two turing machines have always the same output for the identical input. Since you can't even decide whether a tm will halt on the input, I don't see how it should be possible to decide whether both halt AND output the same result...