Continuity at a point vs. interval—contradicton or not?

Question

Let $f(x)=\lfloor x \rfloor $ and imagine posing the following questions.

1. Is $f(x)$ continuous at $x=0$?
2. Is $f(x)$ continuous on $[0,1)$?

For the first question, since $\displaystyle \lim_{x\rightarrow 0} f(x)$ does not exist, we must answer no.

For the second question, since $\displaystyle \lim_{x\rightarrow 0^+} f(x)$ exists and $\forall a \in (0,1) : \displaystyle \lim_{x\rightarrow a} f(x) = f(a)$, we should answer yes.

These are the answers to these two questions based on my understanding of what it means to be continuous at a point and what it means to be continuous on a(n) (closed) interval.

However, in retrospect, this seems bizarre to me given that we are saying that $f(x)$ is not continuous at $0$, while $f(x)$ is continuous on $[0,1)$ even though $0 \in [0,1)$. Is this really the case?

Answer 1

This might be easier to understand using the open set definition of continuity; i.e., a function $f(x) : X \to Y$, where $X$ and $Y$ are topological spaces endowed with specified topologies, is continuous if $f^{-1}(U)$ is open in $X$ for every open set $U$ in $Y$. The upshot here is that continuity of a function is dependent on its domain. In the first case, the domain of $f$ is $\mathbb{R}$, while in the second case it is $[0, 1)$. In $[0, 1)$ with the subspace topology, sets of the form $[0, a)$, $a < 1$ are open.

Answer 2

The reason for the seeming bizarritude is simply that you are suppressing the domains of the functions at issue, and combining that with ambiguous wording. Once you pay attention to domains and use unambiguous language, the problem goes away.

In the first instance, you are considering the function $f$ with domain $\mathbb R$ and asking if that function $f : \mathbb R \to \mathbb R$ is continuous at $0$, and the answer is no.

In the second instance, you are restricting $f$ to have domain $[0,1)$, and asking if that restricted function $f : [0,1) \to \mathbb R$ is continuous at $0$, and the answer is yes.