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Cauchy condition for limit of function
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(Theorem)
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A real function $f$ has the limit $\displaystyle\lim_{x\to x_0}f(x)$ if and only if for every positive number $\varepsilon$ there exists another positive number $\delta(\varepsilon)$ satisfying $$|f(u)-f(v)| < \varepsilon\quad \mbox{when}\quad 0 < |u-x_0| < \delta(\varepsilon)\;\;\mbox{and}\;\;0 < |v-x_0| < \delta(\varepsilon).$$
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- . . : . I . ``''. (1970).
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"Cauchy condition for limit of function" is owned by pahio.
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Cross-references: number, positive, limit, real function
This is version 4 of Cauchy condition for limit of function, born on 2007-12-18, modified 2008-01-27.
Object id is 10147, canonical name is CauchyConditionForLimitOfFunction.
Accessed 1577 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) | | | 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions) | | | 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability) |
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Pending Errata and Addenda
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