Why are strict inequalities used in the definition of continuity?

[Bleys]

I'm just wondering why in the definition of continuity, limit of function, or even just limit of a sequence, the inequality signs are strict? What would happen if you only required that
$$| f(x) - L |\leq \epsilon$$. Or that $$|x-a|\leq \delta$$.

 

[mathman]

Without giving it much thought, I suspect it doesn't matter whether the inequalities are strict or not.

 

[Bleys]

So it was simply decided that the definition had this: it's just convention then? There is no special case that required them to be this way?