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The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator. In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator.
Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
Rational curves are subdivided according to the degree of the polynomial. Degree 1 Line ... Cardiac function curve; Dose–response curve; Growth curve (biology)
The graph of any cubic function is similar to such a curve. The graph of a cubic function is a cubic curve, though many cubic curves are not graphs of functions. Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of ...