Continuity of the function Is the sequence a continuous function on the set of natural numbers?
My book on complex numbers insists that for the function to be continuous, the limit at a point must exist, which, of course, makes sense. But for the last statement to be true, they say that the function must be defined on all points of delta-neighborhood of that point, which kills dead possibility for the function to be continuous at the isolated point of domain of definition. At the same time, my book on math analysis says that it is possible.
Could anybody help me with it?
 A: This depends on the definition of continuity one uses. If one uses the “continuity with limits” approach, then a function can't be continuous on isolated points of its domain.
To the contrary, many people adopt the following definition:

Let $f$ be a real function defined on the set $D\subseteq\mathbb{R}$ and let $x_0\in D$. Then $f$ is continuous at $x_0$ if, for every $\varepsilon>0$, there exists $\delta>0$ such that, for all $x\in D$ with $|x-x_0|<\delta$, the inequality
   $$|f(x)-f(x_0)|<\varepsilon$$
   holds.

With this definition any function is continuous at every isolated point of its domain, because, for a small enough $\delta>0$, the only point $x$ satisfying “$x\in D$ and $|x-x_0|<\delta$” is $x_0$ itself.
When the point $x_0$ is such that an open interval centered at $x_0$ is contained in $D$, then $f$ is continuous at $x_0$ if and only if $f(x_0)=\lim_{x\to x_0}f(x)$ (just prove it, it's easy).
Your math analysis book probably adopts the above definition and not the “with limits” one. With this definition, the restriction of a continuous function to a subset of its domain is automatically continuous.
A: As every definition, the definition of continuity applies to functions, which in turn are object defined by their domain, as well as by their "action". The author of your book decided that it is not sufficiently interesting to investigate whether functions $\mathbb{Z} \to \mathbb{R}$ are continuous or not.
This is not a mistake, but you should be aware that there are ideas, like that of continuity, that have a universal definition among mathematicians. 
Here in Italy the authors of calculus books declare that any function is continuous (by convention) at isolated points of its domain of definition. This agrees with the topological definition of continuity, which is equivalent to that given by @egreg in his answer.
