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Continuity (mathematics)

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In mathematics, a continuous function is one in which "small" changes in the input produce "small" changes in the output. If small changes in the input can produce abrupt changes in the output, the function is said to be discontinuous (or to have a discontinuity).

As an example, consider the function h(t) which describes the height of a growing child at time t. This function is continuous (unless the child's legs were amputated). As another example, if T(x) denotes the air temperature at height x, then this function is also continuous. In fact, there is the dictum 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if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

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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 plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve 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 point c if the following three requirements are satisfied:

  • f(c) must be defined (i.e. c must be an element of the domain of f)
  • The limit of f(x), as x approaches c, must exist
  • The limit of f(x), as x approaches c, must equal f(c)

We call the function everywhere continuous, or simply continuous, if it is continuous at every point of its domain.

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 positive number ε however small, there exists some positive number δ such that for all x 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 teeny 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 teeny the f(x) neighborhood is.

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.
  • 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.

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 1m 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.

Continuous functions between metric spaces

Now consider a function f from one metric space (X, dX) to another metric space (Y, dY). 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 dX(x, c) < δ will also satisfy dY(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 and only if for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = 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 (xn) in X with limit c, the sequence (f(xn)) is a Cauchy sequence. Continuous functions transform convergent sequences into Cauchy sequences.

Continuous functions between topological spaces

More generally still, we can define continuity for functions between topological spaces. Suppose f is a function from a topological space X into a topological space Y. Then f is said to be continuous at a point x0 in X if for every neighborhood V of f(x0) there is a neighborhood U of x0 such that f(U) is a subset of V. If f is continuous at every point of X, then it is simply said to be continuous. It turns out that a function is continuous if and only if the pre-image of every open set is open, and so this is often used as the definition of continuity.

Some facts about continuous maps between topological spaces:

  • If f : XY and g : YZ are continuous, then so is the composition g o f : XZ.
  • If f : XY is continuous and

A continuous map that is bijective such that its inverse map[?] is also continuous is called a homeomorphism.

This notion of continuity is indeed a generalization of the concept for metric spaces: if X and Y are metric spaces and f : XY is a function, then f is continuous as a map between metric spaces if and only if it is continuous as a map between topological spaces.


See also:

References



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