Lateral limits of an endpoint of the interval. Imagine we have a the domain $D=[d_1,d_2]$ of a continuous function $f$.
The definition of right limit I'm using is the following:
$$\lim_{x\rightarrow a^+}f(x)=b \Leftrightarrow \forall_{\epsilon}\exists_{\delta}\forall_{x}(x\in D \ \cap \ ]a,a+\delta[\ \Rightarrow \ f(x) \in N_{\epsilon}(b) )$$, where $N_{\epsilon}(b)$ is the neighbourhood of length $2\epsilon$ at point $b$. We define similarly a left limit.
If I pick  point $a=d_2$, which is a limit point(<=> accumulation point), then the implication is vacuously true, for any value $b$...
Then how can I say that there's no $\displaystyle \lim_{x\rightarrow d_2^+}f(x)$? 
Or when I say that $\lim_{x\rightarrow d_2^-}f(x)=\lim_{x\rightarrow d_2^+}f(x) \Leftrightarrow \lim_{x\rightarrow d_2}f(x) \text{ exists }$, it's valid only for interior points?
Thanks.

I think I get what Fujisaki is talking about. My definition of right limit is incomplete. I should have demanded, right at the begining of the definition, that $a$ be an adherent point to the set $D \cap ]a,+\infty[$, otherwise we get this problem, since $[b,a] \cap ]a,a+\delta[=\emptyset$. 
