Set of functions computable in polynomial time I write paper and I want to distinguish between the class of decision problems which can be decided in polynomial time and the class functions which can be computed in polynomial time. 
The first is just written as $\mathrm{P}$ or $\mathrm{PTIME}$, right? Can the latter be denoted as $m^{\mathcal{O}(1)}$ or is this a confusing definition?
The notation $m^{O(1)}$  just denotes the set of functions of polynomial growth rate, so that would be indeed confusing. The class of functions computable in polynomial time is usually denoted $\mathsf{FP}$.