Complexity of functions evolving in an evolution process are expected to be limited by the time length of an evolution process among other factors. This paper outlines a general method of deriving function-complexity limitations based on mathematical statistics and independent from details of a biological or genetic mechanism of the evolution of the function. Limitations on the emergence of life are derived, these limitations indicate a possibility of a very fast evolution and are consistent with ?RNA world? hypothesis. The discussed method is general and can be used to characterize evolution of more specific biological organism functions and relate functions to genetic structures. The derived general limitations indicate that a co-evolution of multiple functions and species could be a slow process, whereas an evolution of a specific function might proceed very fast, so that no trace of intermediate forms (species) is preserved in fossil records of phenotype or DNA structure; this is consistent with a picture of ?punctuated equilibrium?.