section_path: In calculus/Real function
section_name: Real function
paragraph_text: Real function
  
  A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.
  
  The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.
  
  Functions enjoy pointwise operations, that is, if f and g are functions, their sum, difference and product are functions defined by
  
  The domains of the resulting functions are the intersection of the domains of f and g. The quotient of two functions is defined similarly by
  
  but the domain of the resulting function is obtained by removing the zeros of g from the intersection of the domains of f and g.
  
  The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function x ↦ 1 x, {\displaystyle x\mapsto {\frac {1}{x}},} whose graph is a hyperbola, and whose domain is the whole real line except for 0.
  
  The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that has the original function as a derivative. For example, the function x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for x = 1, is a differentiable function called the natural logarithm.
  
  A real function f is monotonic in an interval if the sign of f (x) − f (y) x − y {\displaystyle {\frac {f(x)-f(y)}{x-y}}} does not depend of the choice of x and y in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function f is monotonic in an interval I, it has an inverse function, which is a real function with domain f (I) and image I. This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.
  
  Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation
  
  such that