Continuous function

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In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the output is not defined), the function is said to be discontinuous (or to have a discontinuity). The context in this entry is real-valued functions on the real domain or on topological or metric spaces other than the complex numbers; for complex-valued functions see complex analysis. The notable difference in approach is that in the present context, the points in the domain that would be regarded as singularities (points of discontinuity) in the complex domain are usually assumed to be absent, or they are explicitly excluded, so as to leave a function that is continuous on a disconnected real domain.

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if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.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.

Real-valued continuous functions

Suppose we have a function that maps real numbers to real numbers and whose domain is 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 when the following two 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 and be equal to f(c). (If the point c in the domain of f is not an accumulation point of the domain, then this condition is vacuously true, since x cannot 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.)

We call the function everywhere continuous, or simply continuous, if it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset.

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.

Alternatively written: Given I,D\subset {\mathbb  {R}} (that is, I and D are subsets of the real numbers), continuity of f:I\to D (read f maps I into D) at c\in {\mathbb  {R}} means that for all \varepsilon >0 there exists a \delta >0 such that |x-c|<\delta and x\in I imply that |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(x) is then continuous at c.

Heine definition of continuity

The following definition of continuity is due to Heine.

A real function f is continuous if for any sequence (x_{n}) such that
\lim \limits _{{n\to \infty }}x_{n}=x_{0},
it holds that
\lim \limits _{{n\to \infty }}f(x_{n})=f(x_{0}).
(We assume that all points x_{n}, x_{0} belong to the domain of f.)

One can say briefly, that a function is continuous if and only if it preserves limits.

Cauchy's and Heine's definition of continuity are equivalent. The usual (easier) proof makes use of the axiom of choice, but in the case of real functions it was proved by Wacław Sierpiński that the axiom of choice is not actually needed. [1]

In more general setting of topological spaces, the concept analogous to Heine definition of continuity is called sequential continuity. In general, sequential continuity is not equivalent to the analogue of Cauchy continuity, which is just called continuity (see continuity (topology) for details).

Examples

  • All polynomials are continuous.
  • If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions f(x)=1/x and g(x)=(sin x)/x. Neither function is defined at x=0, so each has domain R\{0}, and each function is continuous. The question of continuity at x=0 does not arise, since it is not in the domain. The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of g at 0 is 1, g can be extended continuously to R by defining its value at 0 to be 1. A point in the domain that can filled in so that the resulting function is continuous is called a removable singularity. Whether this can be done is not the same as continuity.
  • The rational functions, exponential functions, logarithms, square root function, trigonometric functions and absolute value function are 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 this type of discontinuity as a sudden jump in function values.
  • Another example of a discontinuous function is the sign function.
  • A more complicated example of a discontinuous function is the popcorn function.

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 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 been 1.25m.

As a consequence, if f 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] (or any closed and bounded set) 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 of the minimum of f. (Note that these statements are false if our function is defined on an open interval (a,b) (or any set that is not both closed and bounded). 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 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

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

See also

References

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