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Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $f : X \to Y$ is said to be continuous at $x_0$ if for every $\varepsilon > 0$, there is $\delta > 0$ such that $d_Y(f(x_0), f(x)) < \delta$ whenever $d_X(x_0, x) < \varepsilon$. A map $f : X \to Y$ is said to be continuous if it is continuous at $x_0$ for all $x_0 \in X$.

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A map $f : X \to Y$ is said to be continuous if $U \in \tau_Y$ implies that $f^{-1}(U) \in \tau_X$.

In the case of metric spaces, the metric induces a topology, and the two notions of continuity coincide.

Continuity are necessary conditions for intermediate value theorem, extreme value theorem, as well as differentiability.

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