continuous

A continuous fluid is one in which the mean free path of the fluid is very much less than the characteristic length of the observation.

A function between topological spaces is said to be continuous if the preimage of every open set is an open set.

A metric space is a very special case of a topological space, and the definition of a continuous function between 2 metric spaces is equivalent to the following:
A function f:X→Y between metric spaces X and Y is continuous at a point x∈X iff for all positive ε there exists a positive δ such that for any x', if d(x,x') < δ then d(f(x),f(x')) < ε. That is, if we stay sufficiently close to x, then our image through f stays close to f(x). If f is continuous on all points of a set U, then f is called continuous on U. And if f is continuous on all points of X, f is called simply continuous.

---

In non-standard analysis, we have the following definition:

A (standard!) function f:X→Y between metric spaces X and Y is continuous at a (standard!) point x∈X iff whenever the standard approximation of a (possibly non-standard) x' is x, the standard approximation of f(x') is f(x).

Continuity on a set continues to be defined as before.

Note that we have fewer quantifiers in the definition, making it possibly simpler. We also only need to consider an "infinitesimally small" neighbourhood of x, rather than the "arbitrarily small" neighbourhoods of x required in the "standard" definition.

Probably the simplest way to understand the concept of a continuous function is to think of real functions of a real variable. Something like x²
Here, the word continuous means that the graph of a function should have no breaks. Note that continuity is not a smoothness property. |x| is a continuos function but it has a wedge at the origin. The step function is discontinuous. 1/x is discontinuous at zero, because on one side of zero it goes off to +infinity and on the other side it goes of to -infinity.

Now lets consider a function of two variables. Here the function would trace out a surface. Again, we say that it is continuous if the surface has no breaks in it.

All this thing about epsilons and deltas essentially means that if the function has one value at a point, and then if I move slightly away from that point, the value of the function should not change drastically!

A tense use in many languages to describe an action that, in reference to a fixed point in time, is being carried out and hasn't ended. Also known as the progressive or imperfect tense (as opposed to perfect tense, to denote a completed action).

In English, it is commonly used for present tense. For example:
The cat is playing with the mouse
It is like drinking cleaning fluid with orange juice

It is occasionally used in the past tense to describe a process that occured earlier:
I was sleeping just before you woke me
Japan's economy was developing quickly in the 1960s

And it is even more rarely used for the future:
Next year I will be travelling to Panama
After the rain the sun will be shining

Most other languages use auxiliary verbs to indicate a progressive action. In Thai Yoo or kam lang (literally 'power' or 'energy') is added to the verb in question that is occuring. In Spanish verbs are modified and follow the auxiliary verbs estar, seguir, continuar or any other verb related to motion. Japanese more simply uses auxiliary verb iru following the main verb in -te form, while Vietnamese only have to worry about sliding in the simple modifier dang.

Con*tin"u*ous (?), a. [L. continuus, fr. continere to hold together. See Continent.]

1.

Without break, cessation, or interruption; without intervening space or time; uninterrupted; unbroken; continual; unceasing; constant; continued; protracted; extended; as, a continuous line of railroad; a continuous current of electricity.

he can hear its continuous murmur. Longfellow.

2. Bot.

Not deviating or varying from uninformity; not interrupted; not joined or articulated.

Continuous brake Railroad, a brake which is attached to each car a train, and can be caused to operate in all the cars simultaneously from a point on any car or on the engine. -- Continuous impost. See Impost.

Syn. -- Continuous, Continual. Continuous is the stronger word, and denotes that the continuity or union of parts is absolute and uninterrupted; as, a continuous sheet of ice; a continuous flow of water or of argument. So Daniel Webster speaks of "a continuous and unbroken strain of the martial airs of England." Continual, in most cases, marks a close and unbroken succession of things, rather than absolute continuity. Thus we speak of continual showers, implying a repetition with occasional interruptions; we speak of a person as liable to continual calls, or as subject to continual applications for aid, etc. See Constant.

 

© Webster 1913.