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Point Equation Open Set Metric Space Limit Function Superset Real Number Euclidean Vector Space Property Inverse Image Direct Image Topological Space

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Continuous

Let and 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 .

In the case where and 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 , the point $f(z) \in Y$ has distance less than $\epsilon$ to .

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space at once). When and are topological spaces, we say is continuous at a point $x \in X$ if, for every open subset $V \subset Y$ containing , there is an open subset $U \subset X$ containing whose image is contained in . Of course, the function $f\colon X \to Y$ is continuous in the first sense if and only if 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 and are metric spaces (e.g., Euclidean spaces), a function 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 is continuous at $a \in X$ if and only if the limit of as $x \to a$ satisfies the equation

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

1	Limit
1	Point
1	Euclidean Vector Space
1	Topological Space
1	Open Set
1	Metric Space
1	Real Number
1	Property
1	Direct Image
1	Function
1	Inverse Image
1	Superset
1	Equation