Continuous functions exercise in Spivak's Calculus This is exercise $9$   in chapter $6$ (on continuous functions) of Calculus by Michael Spivak. 

(a) Suppose $f$ is not continuous at $a$. Prove that for some number $\epsilon>0$ there are numbers $x$ arbitrarily close to $a$ with $|f(x) - f(a)|>\epsilon$. Illustrate graphically.
(b) Conclude that for some number $\epsilon>0$ either there are numbers $x$ arbitrarily close to $a$ with $f(x)<f(a) - \epsilon$ or there are numbers $x$ arbitrarily close to $a$  with $f(x)>f(a) + \epsilon$. 

I am having difficulty with $b)$. Consider $f:\mathbb{R} \to \mathbb{R}$ defined as follows
$f(x) =
\begin{cases}
0  & x \in \mathbb{R}\setminus \mathbb{Q} \\
\frac 12 & x=1\\
1 & \{x \in \mathbb{Q} \ | x \neq 1\}\\ 
\end{cases}$
Note $f$ is not continuous at $x=1$. We will thus take $a=1$. 
I am having difficulty finding the appropriate $\epsilon$ which satisfies the "either/or" requirement. If we take $\epsilon\geq\frac 12$ then $f$ satisfies neither of the two requirements, whereas if we take $0<\epsilon < \frac 12$ it satisfies both. What mistake am I making? 
 A: The statement:

there are numbers $x$ arbitrarily close to $a$ with $f(x)<f(a) -
> \epsilon$ or there are numbers $x$ arbitrarily close to $a$  with
  $f(x)>f(a) + \epsilon$.

allows both possibilities to occur (the use of the words "either" and "or" does not imply that the alternatives are mutually exclusive).
A: The "either/or" does not exclude the possibility of both being true here. It is being used to emphasize that  what is meant is not
$$\exists \varepsilon > 0 \ \forall \delta > 0 \ \exists x \in (a-\delta,a+\delta) \quad [f(x) > f(a) + \varepsilon \ \text{ or } \ f(x) < f(a) -\varepsilon]$$
as in the first part, but rather
$$\exists \varepsilon > 0 \ [\forall \delta > 0 \ \exists x \in (a-\delta,a+\delta) \ \ f(x) > f(a) + \varepsilon \quad \text{ or } \quad \forall \delta > 0 \ \exists x \in (a-\delta,a+\delta) \ \  f(x) < f(a) - \varepsilon].$$
It is really being used as a way of stressing that the quantifiers and connectives are placed differently, with the "or" occurring at a higher level in the structure of the statement.
