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continuous (Definition)

Let $ X$ and $ Y$ be topological spaces. A function $ f\colon X \to Y$ is continuous if, for every open set $ U \subset Y$, the inverse image $ f^{-1}(U)$ is an open subset of $ X$.

In the case where $ X$ and $ Y$ are metric spaces (e.g. Euclidean space, or the space of real numbers), a function $ f\colon X \to Y$ is continuous if and only if for every $ x \in X$ and every real number $ \epsilon > 0$, there exists a real number $ \delta > 0$ such that whenever a point $ z \in X$ has distance less than $ \delta$ to $ x$, the point $ f(z) \in Y$ has distance less than $ \epsilon$ to $ f(x)$.

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space $ X$ at once). When $ X$ and $ Y$ are topological spaces, we say $ f$ is continuous at a point $ x \in X$ if, for every open subset $ V \subset Y$ containing $ f(x)$, there is an open subset $ U \subset X$ containing $ x$ whose image $ f(U)$ is contained in $ V$. Of course, the function $ f\colon X \to Y$ is continuous in the first sense if and only if $ f$ is continuous at every point $ x \in X$ in the second sense (for students who haven't seen this before, proving it is a worthwhile exercise).

In the common case where $ X$ and $ Y$ are metric spaces (e.g., Euclidean spaces), a function $ f$ is continuous at $ x \in X$ if and only if for every real number $ \epsilon > 0$, there exists a real number $ \delta > 0$ satisfying the property that $ d_Y(f(x),f(z)) < \epsilon$ for all $ z \in X$ with $ d_X(x,z) < \delta$. Alternatively, the function $ f$ is continuous at $ a \in X$ if and only if the limit of $ f(x)$ as $ x \to a$ satisfies the equation

$\displaystyle \lim_{x \to a} f(x) = f(a). $



"continuous" is owned by djao.
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See Also: limit

Other names:  continuous function, continuous map, continuous mapping
Also defines:  continuous at

Attachments:
discontinuous (Definition) by mathwizard
restriction of a continuous mapping is continuous (Theorem) by matte
continuity of composition of functions (Result) by bbukh
function continuous at only one point (Example) by Andrea Ambrosio
If $f\colon X\to Y$ is continuous then $f\colon X\to f(X)$ is continuous (Theorem) by matte
composition of continuous mappings is continuous (Theorem) by mathcam
continuity is preserved when codomain is extended (Theorem) by matte
gluing together continuous functions (Theorem) by yark
equivalent formulations for continuity (Theorem) by matte
continuity of natural power (Theorem) by pahio
sequentially continuous (Definition) by ehremo
continuity of sine and cosine (Theorem) by pahio
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Cross-references: equation, limit, property, contained, image, distance, point, real numbers, Euclidean space, metric spaces, inverse image, open set, function, topological spaces
There are 589 references to this entry.

This is version 7 of continuous, born on 2001-10-21, modified 2006-10-22.
Object id is 439, canonical name is Continuous.
Accessed 34628 times total.

Classification:
AMS MSC: 54C05 (General topology :: Maps and general types of spaces defined by maps :: Continuous maps)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
Pending Errata and Addenda
None.
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Discussion
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inverse image of _any_ U in Y? by four on 2008-04-15 13:18:23
I think there is some sort of subtlety I'm missing, because it seems like the following example would be non-continuous under the current definition:

X = [0,1]
Y = Real numbers
f(x) = 0 (constant for all x in [0,1]).

Then if U = (-1,1), U is open and in Y, but yet F^-1(U) = {0} which is not open.

A constant function is not continuous!? What is wrong?
[ reply | up ]
non-standard definition by rspuzio on 2004-10-05 15:01:32
I agree completely with you that the non-standard definition is not shorter or simpler once you take into account the fact that, to use it you first have to define non-standard numbers. To do that either reequires ultrafilers or model theory, and both of these are advanced topics.

However, I still thiink it would be a good idea to add the non-standard definition. My reason for saying this is that, if this is supposed to be an encyclopaedia, it should try to include everything, easy or hard. Maybe you could add a sentence like "In non-standard analysis, a function is defined to be continuous if ...". One of these days (if someone else doesn't do it first), I plan to add something about non-standard analysis. Probably, I'll lump it together with p-adic analysis in a topic entry on non-Archimedian analysis.
[ reply | up ]
metric spaces by vitriol on 2002-02-21 04:36:51
would it be possible to have a more elementary definition that doesn't involve metric spaces ?
[ reply | up ]
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