Let $\displaystyle

f:A \subset \mathbb{R} \to \mathbb{R}

$ and $\displaystyle r_n > 0$

a sequence such that $\displaystyle

r_n \to 0

$

Prove that f is continuos in $\displaystyle \overline x $

if and only if

$\displaystyle

s_n = \sup \left\{ {\left| {f\left( x \right) - f\left( {\overline x } \right)} \right|:\,\left| {x - \overline x } \right| \leqslant r_n } \right\}

$ converges to zero

thanks!