Cauchy condition for limit of function

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).

References

1 Л. Д. Кудрявцев: Математический анализ. I том. Издательство ‘‘Высшая школа’’. Москва (1970).

Title	Cauchy condition for limit of function
Canonical name	CauchyConditionForLimitOfFunction
Date of creation	2013-03-22 17:42:18
Last modified on	2013-03-22 17:42:18
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26B12
Classification	msc 26A06
Classification	msc 54E35
Synonym	necessary and sufficient condition of limit
Related topic	Complete
Related topic	CauchyCriterionForTheExistenceOfALimitOfAFunction