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).