An algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis is presented in this paper. Main result: For any function \(f\) over the real numbers being 2-times continuously differentiable defined on a product of compact interval with rational endpoints, \(f\) is elementarily computable in the sense of recursive analysis if and only if \(f\) belongs to the smallest class of functions containing some basic functions and closed by composition, linear integration and a simple limit schema, and \(f\) is elementarily computable in the sense of recursive analysis in level \(n\geq 3\) of the Grzegorczyk hierarchy if and only if \(f\) belongs to the smallest class of functions containing some (other) basic functions and closed by composition, linear integration and a simple limit schema.