This may be just a grammatical question.
I have seen the following two representations:
- This function is continuous in $D.$
- This function is continuous on $D.$
Is there any difference between them?
I thought that "in $D$" means the inside of $D$
while "on $D$" contains its boundary, but
I have faced many exceptions.
Could anyone explain the difference?
I would appreciate if you could tell me
other cases where this kind of difference matters.
Best Answer
Consider the function $f: \mathbb R \to \mathbb R $ defined by $f(x)=1$ for $x \in [0,1]$ and $f(x)=0$ for $x \in \mathbb R -[0,1]$. The function is not continuous but the restriction $f:[0,1] \to \mathbb R $ is just the constant function so is continuous. For the subset of the domain $D=[0,1]$ we say $f$ is continuous on $D$.
Consider the function $f(C,D) = \lfloor C \rfloor + D$ of variables $C$ and $D$. The function is not continuous but for each fixed $C_0$ the function $D \mapsto f(C_0,D) = \lfloor C_0 \rfloor + D\ $is continuous. In this case we say $f$ is continuous in $D$.
I have never seen these terms referring to the boundary. More common terms are continuous on $\boldsymbol D$ or continuous on the interior of $\boldsymbol D$. Another common thing to say is differentiable/holomorphic on the interior of $\boldsymbol D$ and continuous on the boundary.