The continuity of a function defined on disjoint union.

Question

Let $X\subset \mathbb R$ and $f:X\to \mathbb R$ be a function.

I want to show

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If $A,B\subset \mathbb R$ satisfy $$X=A\cup B, A\cap B=\emptyset,$$ then

$f$ is continuous $\iff$ $f|_A$, $f|_B$ are both continuous.

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I did $\Rightarrow$. So I have to show opposite.

Suppose $f|_A$, $f|_B$ are both continuous.

To show the continuity of $f:X\to \mathbb R$, I'll show $$\forall O\in \mathcal O_{\mathbb R}\ ; \ f^{-1}(O)\in \mathcal O_{\mathbb R}^X,$$ where $\mathcal O_{\mathbb R}$ is Euclidean topology and $\mathcal O_{\mathbb R}^X$ is relative topology, i.e., $\mathcal O_{\mathbb R}^X=\{ X\cap O \mid O\in \mathcal O_{\mathbb R} \}.$

I have $$f^{-1}(O)=(f|_A)^{-1}(O)\cup (f|_B)^{-1}(O).$$

And from the continuity of $f|_A$ and $f|_B$, I have $$(f|_A)^{-1}(O)\in \mathcal O_{\mathbb R}^A,\ (f|_B)^{-1}(O)\in \mathcal O_{\mathbb R}^B.$$ So there exists $O_1, O_2\in \mathcal O_{\mathbb R}$ s.t. $$(f|_A)^{-1}(O)=A\cap O_1,\ (f|_B)^{-1}(O)=B\cap O_2.$$

Then, \begin{align} f^{-1}(O) &=(f|_A)^{-1}(O)\cup (f|_B)^{-1}(O)\\ &=(A\cap O_1) \cup (B\cap O_2) \end{align}

So, to show $f^{-1}(O)\in \mathcal O_{\mathbb R}^X$, I want to say $$(A\cap O_1) \cup (B\cap O_2) =X\cap [\mathrm{an \ element \ of\ } \mathcal O_{\mathbb R}]. \ \cdots (\ast) $$

But I'm stacked here. I don't know how I should show $(\ast)$.

Answer 1

$\Longleftarrow$ is false. In his comment Sassatelli Giulio gave the counterexample $X = \mathbb R, A = (-\infty,0), B= [0,\infty), f(x) = \begin{cases} 1 & x \in A \\ 0 & x \in B\end{cases}$.

But, as Paul Frost comments, perhaps you misunderstood the concept of disjoint union in topology. A space $X$ is the disjoint union of subspaces $A, B$ if

1. $A \cup B = X$.
2. $A \cap B = \emptyset$.
3. $A, B$ are open in $X$.

Note that 3. is equivalent to the requirement that $A, B$ are closed in $X$.

In your question you consider the case $X \subset \mathbb R$. Condition 3. does not mean that $A, B$ are open (or closed) in $\mathbb R$. An example is $X = (0,1] \cup [2,3), A = (0,1], B = [2,3)$.

Now forget that $X \subset \mathbb R$. We only need to know that it is a topological space.

To show that $f$ is continuous, we have to show that for $f^{-1}(U)$ is open in $X$ for each open $U \subset \mathbb R$. We know that $(f \mid_A)^{-1}(U) = f^{-1}(U) \cap A$ is open in $A$ and $(f \mid_B)^{-1}(U) = f^{-1}(U) \cap B$ is open in $B$. But $A$ and $B$ are open in $X$, thus also $f^{-1}(U) \cap A$ and $f^{-1}(U) \cap B$ are open in $X$. Therefore $f^{-1}(U) = f^{-1}(U) \cap A \cup f^{-1}(U) \cap B$ is open in $X$.