Continuous function
In mathematics, a continuous function is one 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, 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 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if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.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.
Table of contents |
2 Continuous functions between metric spaces 3 Continuous functions between topological spaces 4 See also 5 References |
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 when the following two requirements are satisfied:
- f(c) must be defined (i.e. c must be an element of the domain of f).
- If c is an accumulation point of the domain, then the limit of f(x) as x approaches c must exist and be equal to f(c).
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.
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.
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, 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.
Continuous functions between topological spaces
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.
See also
- uniform continuity
- bounded linear operator
- absolute continuity
- semicontinuity
- normal function
- equicontinuity
- continuous functor
- Lipschitz continuity
References
- Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)