I found the following problem a little tricky and I'd be glad if I could get some direction (just a hint):
Let f be a function defined around $x_o$. For every $\epsilon>0$ there's some $\delta>0$ such that if $0<|x-x_0|<\delta$ and $0<|y-x_0|<\delta$ then $|f(x)-f(y)|<\epsilon$.
What's needed to be proven is that $\lim_{x\to x_0}f(x)$ exists.
A big thanks in advance.
P.S. I've looked at the opposite question, where you need to prove what is given here, and you are given what is needed to prove here. It was easier as you could just add $L$ & $-L$ inside the absolute value and then just use the triangle inequality and play with the epsilon, but proving the opposite (what I asked above) appears to be trickier.