About the definition of one sided limits In the wikipedia page about one sided limits at: http://en.wikipedia.org/wiki/One-sided_limit the rigorous definition of the right sided limit is given as:
$$\forall \epsilon >0, \exists \delta>0, \forall x\in I (0<x-a<\delta)\implies |f(x)-L|<\epsilon$$
It is similar for the left sided one. In the above definition $I$ is defined as an interval within the domain of $f$.
What disturbs me about this definition is that it can allow a weird thing as the following:

In this sketch the function $f$ has the interval $[a,b]$ as its domain. The function $f$ is constant with $f(x)=y'$. Now I think that the above definition of the right sided limit allows the evaluation of the limit $\lim_{x \to c^{-}}f(x) = y'$. Here, for each $\epsilon > 0$, we have for all $x \in [a,b)$ and for $\delta = b - c$ that it is $0 < x - c < \delta \implies |f(x) - y'| < \epsilon $. But this seems illogical as $c$ is not even contained in $f$'s domain. I know that $f$ can be undefined at $c$ in the case of the regular, double sided limit $\lim_{x \to c} f(x)$ but it needs to be contained in the domain of $f$. Since no such containment condition has been given for the definition of the one sided limit, it allows such a weird limit to be evaluated.
Am I wrong with my assessment here? Or is it natural that one sided limits allow such situations?

Your observation is correct, but this is just inaccuracy in the definition. The one sided limit at a point $a$ can only exist if $a$ is in the closure of the domain of $f$, i.e. if for any $\delta>0$ there is a point $b$ in the domain of $f$ with $0<b-a<\delta$ (or $0<a-b<\delta$, when the limit is from the other side).
