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Let $f: A \rightarrow \mathbb{R}$ be a real function defined on a subset $A$ of the real line. We say that $f$ is uniformly continuous if, given an arbitrary small $\varepsilon$, there exists a $\delta$ such that whenever two points differ for less than $\delta$, they are mapped in points which differ for less than $\varepsilon$. In symbols:
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Let $f: A \rightarrow \mathbb{R}$ be a real function defined on a subset $A$ of the real line. We say that $f$ is \emph{uniformly continuous} in $A$ if
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| \[ \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in A\ |x-y| < \delta \Rightarrow |f(x)-f(y)| < \varepsilon. \] |
\[ \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in A\ |x-y| < \delta \Rightarrow |f(x)-f(y)| < \varepsilon. \] |
| Uniform continuity is stronger than continuity, in fact the $\delta$ here does not depend on $x$ and $y$. Thus every uniformly continuous function is also continuous, while the converse does not always hold. For instance, the function $f: [0,+\infty[ \rightarrow \mathbb{R}$ defined by $f(x) = 1/x$ is continuous in its domain, but not uniformly. |
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| A more general definition of uniform continuity applies to functions between metric spaces (there are even more general environments for uniformly continuous functions, i.e. uniform spaces). |
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| Given a function $f: X \rightarrow Y$, where $X$ and $Y$ are metric spaces with distances $d_X$ and $d_Y$, we say that $f$ is uniformly continuous if |
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| \[ \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in A\ d_X(x,y) < \delta \Rightarrow d_Y(f(x),f(y)) < \varepsilon. \] |
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| Uniformly continuous functions have the property to map Cauchy sequences in Cauchy sequences and to preserve uniform convergence of function sequences. |
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