The following are equivalent:
F: A→B is a constant function. For all functions g, h: c → a, fog = foh ("o" stands for composite function). ? The composition of f with any other function is still a constant function. The first description of the constant function given above is the nature of the excitation and definition of the more general concept of constant morphism in category theory
According to the definition, the derivative function of a function measures the relationship between the change of independent variables and the change of the function. Then we can get that because the value of a constant function is constant, its derivative function is zero.
For example:
If f is a real number function defined in a certain interval and the variable is a real number, then if and only if the derivative constant of f is zero, f is a constant. For functions between pre-ordered sets, constant functions are order-preserving and inverse; On the other hand, if F is both order-preserving and inverse, if its domain is lattice, then F must be a constant function.
Other properties of constant functions include:
Any constant function with the same domain and cotangent is idempotent. Constants in any topological space are continuous. In a connected set, f is a local constant if and only if it is a constant.
For example, when proving Rolle's theorem, for the first case, it is deduced that m = m and F (x) = constant. 2。 According to the definition of extreme value of function (defined in advanced mathematics of Tongji University Edition), constant function has no extreme value.
Because in the definition of maximum (minimum), the maximum point (minimum point) needs a neighborhood, so that the function value of any point in the neighborhood is less than (greater than) the function value of the maximum point (minimum point).
So any constant function does not conform to the definition of extreme value.