Let $f:A\to\mathbb{R}$ and $c$ a limit point of $A$ (there is a sequence of elements of $A\setminus \{c\}$ that converges to $c$). Consider $\delta>0$. The oscillation of $f$ in the neighbourhood $V_\delta (c)$ is defined as the number $w(f,\delta):=sup\{\left|f(x)-f(y)\right|:x,y\in V_\delta (c)\cap A\}$. Show that if $\displaystyle\lim_{x\to c}f(x)=L\in\mathbb{R}$, then for every $\varepsilon >0$, there is a $\delta>0$ such that $w(f,\delta)<\varepsilon$
I was thinking about adding and subtracting $L$ in the term $\left|f(x)-f(y)\right|$ and using a nighbourhod that works for $\varepsilon/2$ in the definition of limit. However, I didn't get what I wanted.