Elementary Theories of Continuous Functions Spaces

Method of generalized interpretations with its applicability for the proving of theories decidability was studied. By this method the decidability of the continuous functions theory from $\mathbb{R}$ to $\mathbb{R}$ lattice and from $\mathbb{R}^n$ to $\mathbb{R}$ lattice has been proven. The undecidability of theory of continuous functions structure with additional unary predicate which distinguishes constants has been proven as well. This study demonstrated that the new method may be considered as a powerful tool for establishing the decidability of elementary theories.