Continuous Function Real valued Continuous Functions

As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous (unless the flower is cut). As another example, if T(x) denotes the air temperature at height x, then this function is also continuous. In fact, there is 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if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

There are also some more special usages of continuity in some mathematical disciplines. Probably the most common one, found in topology, is described in the article on continuity (topology). In order theory, especially in domain theory, one considers a notion derived from this basic definition, which is known as Scott continuity.

1 Real-valued continuous functions

Suppose we have a function that maps real numbers to real numbers and is defined on some interval, like the three functions h, T and M from above. Such a function can be represented by a graph in the Cartesian planeCartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. This work was influential to the development of analytic geometry, calculus, and cartography. The idea; the function is continuous if, roughly speaking, the graph is a single unbroken curveThis article is about the term used in mathematics. There is also a magazine called Curve. Metric geometry Geometry Topology General topology In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and c with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper.

To be more precise, we say that the function f is continuous at some pointA spatial point is an entity with a location in space but no extent ( volume, area or length). In geometry, a point therefore captures the notion of location no further information is captured. Points are used in the basic language of geometry, physics, v c when the following two requirements are satisfied:

We call the function everywhere continuous, or simply continuous, if it is continuous at every point of its domainSet theory In mathematics, the domain of a function is the set of all input values to the function. Given a function , the set A is called the domain or domain of definition of f. The set of all values in the codomain that f maps to is called the range of. More generally, we say that a function is continuous on some subsetIf X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes X; Y ⊇ X. Every set Y is a subset of itself. A subset of Y which is not equa of its domain if it is continuous at every point of that subset.

1.1 Epsilon-delta definition

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 (and only 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) will satisfy f(c) - ε < f(x) < f(c) + ε. 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.

1.2 Examples

All polynomials are continuous, and so are the exponential functions, logarithms, square root function and trigonometric functions.
The absolute value function is also continuous.
The real function f of non-zero real numbers such that f(x) = 1/x is continuous. However, if the function is extended by assigning some value to f(0), the extension will not be continuous.
An example of a discontinuous function is the function f defined by f(x) = 1 if x > 0, f(x) = 0 if x ≤ 0. Pick for instance ε = 1/2. There is no δ-neighborhood around x=0 that will force all the f(x) values to be within ε of f(0). Intuitively we can think of a discontinuity as a sudden jump in function values.
Another example of a discontinuous function is the sign function.

1.3 Facts about continuous functions

If two functions f and g are continuous, then f + g and fg are continuous. If g(x) ≠ 0 for all x in the domain, then f/g is also continuous.

The composition f o g of two continuous functions is continuous.

The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states: "If the real-valued function f(x) is continuous on the closed interval [a, b] and k is some number between f(a) and f(b), then there is some number c in [a, b] such that f(c) = k. For example, if a child undergoes continuous growth from 1 m to 1.5m between the ages of 2 years and 6 years, then, at some time between 2 years and 6 years of age, the child's height must have equalled 1.25m.

As a consequence, if f(x) is continuous on [a, b] and f(a) and f(b) differ in sign, then, at some point c, f(c) must equal zero.

If a function f is defined on a closed interval [a,b] and is continuous there, then the function attains its maximum, i.e. there exists c∈[a,b] with f(c) ≥ f(x) for all x∈[a,b]. The same is true for the minimum of f. (Note that these statements are false if our function is defined on an open interval (a,b). Consider for instance the continuous function f(x) = 1/x defined on the open interval (0,1).)

If a function is differentiable at some point c of its domain, then it is also continuous at c. The converse is not true: a function that's continuous at c need not be differentiable there. Consider for instance the absolute value function at c=0.

2 Continuous functions between metric spaces

Now consider a function f from one metric space (X, d_X) to another metric space (Y, d_Y). Then f is continuous at the point c in X if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying d_X(x, c) < δ will also satisfy d_Y(f(x), f(c)) < ε.

This can also be formulated in terms of sequences and limits: the function f is continuous at the point c if for every sequence (x_n) in X with limit lim x_n = c, we have lim f(x_n) = f(c). Continuous functions transform limits into limits.

This latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (x_n) in X with limit c, the sequence (f(x_n)) is a Cauchy sequence. Continuous functions transform convergent sequences into Cauchy sequences.

3 Continuous functions between topological spaces

Main article: continuity (topology)

The above definitions of continuous functions can be generalized to functions from one topological spaces to another in a natural way; a function f : X → Y, where X and Y are topological spaces, is continuous iff for every open set V ⊆ Y, f^-1(V) is open in X.

4 See also

5 References

Visual Calculus by Lawrence S. Husch , University of Tennessee ( 2001)

Calculus Topology General topology

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