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Function. In mathematics, the graph of a function is the set of ordered pairs , where In the common case where and are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve. The graphical representation of the graph of a function is also known as a plot. In the case of functions of two variables ...
Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup ∪ {\displaystyle \cup } (or a straight line like a linear function), while a concave function 's graph is shaped like a cap ∩ {\displaystyle \cap } .
A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.
Floor function. Ceiling function. In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor (x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil (x).
Bounded function. A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics, a function defined on some set with real or complex values is called bounded if the set of its values ...
Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial. Rational functions: A ratio of two polynomials. n th root.
Closed graph property. In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. [1] [2] A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstrass function has historically served the role of a pathological function, being the first published example ...