Why is the empty set considered an interval? What is the definition of an interval and why is the empty set an interval by that definition?
 A: The definition of a real interval is: a subset $I$ of $\Bbb R$ such that 
$$\forall x\in I, \forall y\in I,\forall z\in \Bbb R, x<z<y \implies z\in I$$
Notice that $\forall x\in \emptyset, P(x)$ is always true by definition.
See also vacuous truth on Wikipedia, and these questions on MSE:
Why is predicate "all" as in all(SET) true if the SET is empty?
Why is "for all $x\in\varnothing$, $P(x)$" true, but "there exists $x\in\varnothing$ such that $P(x)$" false?
A: Given an ordered set $(A,\leq)$ an interval is a subset which is convex. Namely $I$ is an interval in $A$ if whenever $a,b\in I$ then for every $x$ such that $a<x<b$, we have that $x\in I$ as well.
In the real numbers, because the order is complete, it follows that every bounded interval has endpoints, and may or may not include them. For example $[0,1]$ is the interval of all numbers from $0$ to $1$, including these two, and $(\sqrt2,42]$ is the interval of all numbers strictly larger than $\sqrt2$ but not strictly larger than $42$.
But nowhere in the books it is said that the endpoints of the interval have to be different from one another. What would be $(0,0)$? It would be all the numbers strictly between $0$ and $0$. But no such numbers exist, so $(0,0)=\varnothing$.
And indeed, whenever $a,b\in\varnothing$, and $x$ lies between $a$ and $b$, it follows that $x\in\varnothing$ as well. If you want to claim that this is false, you need to come up with actual $a,b\in\varnothing$ and $x$ between them which is not an element of $\varnothing$. Coming up with $x$ is easy, but coming up with $a$ and $b$ is impossible. So the definition is satisfied vacuously and $\varnothing$ is an interval.
