What are the requirements for a statement to have a constructive proof? In general when trying to solve an excersise, or construct a proof, I always find myself looking at what strategy should I take to complete the proof. Many times I try to solve the excercise with a constructive proof, and since it is not working out, I end up trying with an absurd-type argument. A couple of days ago, I read a question asking if there was any way to exhibit a constructive proof for a certain theorem regarding continuity, and someone answered that there was no way to give such a proof if working in ZFC. 
My question is, how can you know, if you can give a constructive proof or not? Is there any way to give a constructive proof of this question?
 A: 
Given a theorem, (no one in particular), is there any way to know (not necessarily constructively) if there is a constructive proof for the theorem?

No, there is no such method. 
First, there is no formal notion of "constructive proof" in general - there are just particular constructive proof systems. So the best you could hope for is a way to tell if a statement is provable in some particular constructive system.
But even that is impossible, because of the undecidability of the halting problem. This asks for an algorithm to determine, given a Turing machine, whether the Turing machine will eventually halt after it is started. Now, a Turing machine halts if and only if there is a constructive proof that the machine halts. If it halts, we can list out the execution step by step to prove that it halts -- and the converse also holds. This proof method will work in any sufficiently strong constructive proof system.  
So, if there was a way to tell whether a given theorem had a constructive proof, in a sufficiently strong constructive system, then there would be a way to tell whether a given Turing machine halts. We know that is not the case. 
