Continuous function Totally Explained

Everything about Continuous Function totally explained

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. An intuitive though imprecise (and inexact) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the blackboard.
Continuity of functions is one of the core concepts of topology, which is treated in full generality in a more advanced article. This introductory article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity.
As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous. In fact, there's a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

Real-valued continuous functions

Suppose we've a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".
To be more precise, we say that the function f is continuous at some point c when the following two requirements are satisfied:

f(c) must be defined (for example c must be an element of the domain of f).
The limit of f(x) as x approaches c must exist and be equal to f(c). (If the point c in the domain of f isn't a limit point of the domain, then this condition is vacuously true, since x can't approach c. Thus, for example, every function whose domain is the set of all integers is continuous, merely for lack of opportunity to be otherwise. However, one doesn't usually talk about continuous functions in this setting.)

We call the function everywhere continuous, or simply continuous, if it's continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it's continuous at every point of that subset. If we simply say that a function is continuous, we usually mean that it's continuous for all real numbers.
The notation C(Ω) or C⁰(Ω) is sometimes used to denote the set of all continuous functions with domain Ω. Similarly, C¹(Ω) is used to denote the set of differentiable functions whose derivative is continuous, C²(Ω) for the twice-differentiable functions whose second derivative is continuous, and so on. In the field of computer graphics, these three levels are sometimes called g0 (continuity of position), g1 (continuity of tangency), and g2 (continuity of curvature). The notation C^{(n, α)}(Ω) occurs in the definition of a more subtle concept, that of Hölder continuity.

Cauchy definition (epsilon-delta)

Without resorting to limits, one can define continuity of real functions as follows.
Again consider a function f that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of f. The function f is said to be continuous at the point c if the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain with c − δ < x < c + δ, the value of f(x) satisfies ť

f(c) - varepsilon < f(x) < f(c) + varepsilon.,

Alternatively written: Given subsets I, D of R, continuity of f : I → D at c ∈ I means that for all ε > 0 there exists a δ > 0 such that for all x ∈ I : ť

| x - c | < delta Rightarrow | f(x) - f(c) | < varepsilon.

This "epsilon-delta definition" of continuity was first given by Cauchy.
More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(c), we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter how small the f(x) neighborhood is; f is then continuous at c.

Heine definition of continuity

The following definition of continuity is due to Heine. ť A real function f is continuous if for any sequence (x_n) such that

:

limlimits_

is open.
However, this definition is often difficult to use directly. Instead, suppose we've a function f from X to Y, where X,Y are topological spaces. We say f is continuous at x for some

x in X

if for any neighborhood V of f(x), there's a neighborhood U of x such that

f(U) subseteq V

. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can always find a U containing x that will map inside it. If f is continuous at every

x in X

, then we simply say f is continuous. In a metric space, it's equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. This leads to the standard ε-δ definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance.
Note, however, that if the target space is Hausdorff, it's still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

Continuous functions between partially ordered sets

In order theory, continuity of a function between posets is Scott continuity. Let X be a complete lattice, then a function f : X → X is continuous if, for each subset Y of X, we've sup f(Y) = f(sup Y).

Continuous binary relation

A binary relation R on A is continuous if R(a, b) whenever there are sequences (a^k)_i and (b^k)_i in A which converge to a and b respectively for which R(a^k, b^k) for all k. Clearly, if one treats R as a characteristic function in three variables, this definition of continuous is identical to that for continuous functions.

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