The Gauss floor function, denoted as [x], is defined as the greatest integer less than or equal to x. For example, [2.7] equals 2, [3] equals 3, and [-1.5] equals -2. This function creates a step-like graph that jumps at each integer value. Elementary functions include polynomials, exponentials, logarithms, and trigonometric functions. Our goal is to prove that the floor function can be expressed as a composition of these elementary functions.
Function composition is a fundamental operation that combines two functions into one. The composition f of g, written as f composed with g of x, equals f of g of x. Elementary functions can be combined through composition, addition, multiplication, and limits to create more complex functions. For example, e to the x squared is the composition of the exponential function with the polynomial x squared. Similarly, sine of natural log of x combines trigonometric and logarithmic functions. This framework allows us to express complex mathematical objects, like the floor function, in terms of simpler elementary components.
The analytical representation method provides our first proof approach. We start with the integral formula: floor of x equals x minus the integral from 0 to 1 of the fractional part of x plus t, dt. The fractional part function can be expressed using a Fourier series as one half minus one over pi times the sum from n equals 1 to infinity of sine of 2 pi n y over n. Substituting this into our integral and evaluating, we obtain floor of x equals x minus one half plus one over pi times the sum of sine terms. This demonstrates that the floor function can indeed be expressed as a composition of elementary functions: polynomials, trigonometric functions, and their infinite series.
The limit-based construction provides our second proof method. We express the floor function as the limit as n approaches infinity of x minus one over n times sine of 2 pi n x over 2 pi. This formula combines elementary functions: the polynomial x, rational functions like one over n, and trigonometric functions like sine. As n increases, the oscillating sine term becomes smaller everywhere except near integer points, where it preserves the discontinuous jumps. The convergence process shows how smooth elementary functions can approximate the step-like behavior of the floor function. This limit construction demonstrates another way to express the floor function as a composition of elementary functions.