$(\Rightarrow):$ $f, \ a\text{'}$da sürekli ve $\epsilon>0$ olsun.
$\left.\begin{array}{rr} \epsilon>0 \\ f, \ a\text{'da sürekli} \end{array}\right\}\Rightarrow (\exists\delta>0)(A\cap (a-\delta,a+\delta)\subseteq f^{-1}[(f(a)-\epsilon,f(a)+\epsilon)])$
$\Rightarrow (\exists\delta>0)(A\cap [(a-\delta,a)\cup (a,a+\delta)]\subseteq f^{-1}[(f(a)-\epsilon,f(a)+\epsilon)])\Big{/}\lim\limits_{x\to a}f(x)=f(a).$
$------------------------------------$
$(\Leftarrow):$ $\lim\limits_{x\to a}f(x)=f(a)$ ve $\epsilon>0$ olsun.
$\left.\begin{array}{rr} \epsilon>0 \\ \lim\limits_{x\to a}f(x)=f(a) \end{array}\right\}\Rightarrow (\exists\delta>0)(A\cap [(a-\delta,a)\cup (a,a+\delta)]\subseteq f^{-1}[(f(a)-\epsilon,f(a)+\epsilon)])$
$\Rightarrow (\exists\delta>0)(A\cap (a-\delta,a+\delta)\subseteq f^{-1}[(f(a)-\epsilon,f(a)+\epsilon)])\Big{/} f, \ a\text{'da sürekli.}$