Prove the limit of a piecewise function? Question is in picture. For some reason this is totally boggling me, I have no idea how to apply the epsilon-delta definition to this. I feel a little lost approaching piecewise functions, I feel like I'm missing a crucial concept and I'd appreciate any help.

Let $f(x)$ be the function whose domain is all $\mathbb R$ given by the rule
  $$f(x) = \begin{cases}
42 & \text{if } x \in \mathbb Z\\
0 & \text{if } x \notin \mathbb Z
\end{cases}$$

Meant to prove that the limit is $0$. I'm pretty confused, does the limit even exist? It seems to me that the limit is $42$ if $x$ is approaching an integer and $0$ if $x$ is approaching any other real number. 
 A: Recall the definition of the limit $\lim\limits_{x \to x_0} f(x) = L$
$$\forall \varepsilon > 0\ \exists \delta > 0 \text{ s.t. } 0 < |x - x_0| < \delta \implies |f(x) - L| < \varepsilon$$
As you can see, the value of $f$ at $x_0$ it's never taken into consideration. The point $x_0$ could be outside $f$'s dominion and the limit wouldn't change. What's important, instead, are the neighborhoods of $x_0$.
We want to prove that the limit is $0$ everywhere. Given $\varepsilon > 0$, we want to find $\delta$ such that
$$0 < |x - x_0| < \delta \implies |f(x)| < \varepsilon$$
Can you do that? Hint: consider $\delta \in (\lfloor x_0\rfloor, \lfloor x_0\rfloor + 1)$.
A: As you approach $x$, the limit is $0$, for any $x$. It doesn't matter whether or not $x$ is an integer because a limit does not care about the value at $x$, it cares about what happens as you approach $x$.
A: In intuitive terms, the limit at a point is the same value as that at the close neighbors *.
For the given $f$, whatever the $x$, integer or not, the value at the close neighbors is $0$ (if the chosen neighborhood contains an integer, you can take a smaller neighborhood).
More specifically, 


*

*if $x$ is an integer, $f=0$ over $(x-1,x+1)$;

*if $x$ is a non-integer, $f=0$ over $(\lfloor x\rfloor,\lceil x\rceil)$.
The value of $f$ at $x$ itself is not taken into account.

*Mind the oversimplification. Usually the values in the neighborhood aren't equal. So the true definition of a limit works with the idea that you can find a neighborhood where the values are as close as you want from each other, i.e. the variations are bounded by an arbitrary tolerance.
