I have an optimization problem, whose answer is a real value, not an integer such as vertex cover and set cover. Therefore, the decision version of my problem is given an input and a real value $r$.
I have been able to reduce an NP-complete problem to my own problem in polynomial time. I also showed that my problem is NP.
Since the input to the decision problem is a real value, is this reduction valid and can I categorize my problem as NP-complete?
Edit: What if the precision of this real number is limited to $\frac{1}{polynomial(n)}$, which means that the solution is a real number with a polynomial precision.