Fractional powers with rational exponent of the monotonous operators of class C¹ on R

The study of functional equation have started after the year of 1700, great mathematicians like Leonard Euler or K. F. Gauss have thought about it.

During periods of time, an entire theory has been developed around this equation, mathematicians all around the world trying to solve more complicated functional equation, like P(f(x))=g(x), where P(x) is a polynomial of n degree with real or complex coefficients.

Along with the evolution of mathematics, R had been replaced at the definition of functions g with more complicated spaces, like , or at the beginning of the 20th century with Hilbert H infinite dimensional spaces or even with Banach spaces.

Moving on, we will solve the functional equation in the most basic case, the one researched ever since 1700.