Introduction to Analysis: Actually Constant I more or less understand the concept of locally constant. If a function, $f(x)$, is locally constant then there is a sufficiently small neighborhood $(a - \delta, a + \delta)$ about $a$ such that $f(x)$ is constant.
However, what do they mean when they say Actually Constant? 
 A: A little topology goes a long way here. The key to your underlying question is that real intervals are connected.
I leave proofs of the following as exercises for the reader. There are several paths you can take to get from Lemma 1 down to Theorem 5. Start with Corollary 2 or Corollary 3. If you took Corollary 2, prove Corollary 3 or Corollary 4. Note: it is probably easier to use Corollary 3 to get to Theorem 5 than Corollary 4.
Let $f\colon X\to Y$.
Lemma 1: $f$ is locally constant iff for each $p\in Y$, $f^{-1}\{p\}$ is open.
Corollary 2: $f$ is locally constant iff $f$ is continuous when $Y$ is given the discrete topology. 
Corollary 3: $f$ is locally constant iff for each $S\subseteq Y$, $f^{-1}[S]$ is open.
Corollary 4: If $f$ is locally constant, then $f$ is continuous, regardless of the topology on $Y$.
Theorem 5: A locally constant function on a connected set is constant.
A: Actually constant means constant over the whole domain.
To find an example of a function (even a continuous one) that is locally constant but not actually constant, consider a non-connected domain such as $(0,1) \cup (2,3)$ and consider the function $f(x)=\lfloor x \rfloor$. Then $f$ is $0$ in $(0,1)$ and $2$ in $(2,3)$. It is not constant but it is locally constant.
