# Real Analysis(continuity)

Prove or give a counterexample. let $f \colon R \to R$ such that $f$ is continuous on $D \subseteq R$ and $f$ is continuous on $S \subseteq R$. Then f is continuous on $D \cup S$. I am kind of puzzled with notation ($f$ is continuous on $D \subseteq R$ and $f$ is continuous on $S \subseteq R$. Then $f$ is continuous on $D \cup S$) Can you give me an example?

What if I consider this example: f(x) = x if x is rational and f(x) = 0 if x is irrational. Then I will have that f is continuous only at 0.( a counterexample)?

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What part of the notation puzzles you? (At the moment, you simply restate verbatim the question.) – Did Mar 8 '12 at 10:31
Suppose $f$ is continuous on $(-1,0]$ and that $f$ is continuous on $(0,1)$. Does $f$ have to be continuous on $(-1,1)$? – Tom Cooney Mar 8 '12 at 10:41
Or maybe do an example like this: $f$ is continuous on $\mathbb R \setminus \{0\}$ and $f$ is continuous on $\{0\}$. – GEdgar Mar 8 '12 at 13:31
There is a semantic problem here that has not been addressed so far: A function $f:\ {\mathbb R}\to{\mathbb R}$ has a set of continuity $C_f:=\{x\in{\mathbb R}\ |\ f\ {\rm is\ continuous\ at}\ x\}$. When $D\subset C_f$ and $S\subset C_f$ then obviously $D\cup S\ \subset C_f$. On the other hand such an $f$ can be restricted to some given subset $D\subset{\mathbb R}$ which is now a topological space in its own right. When $f_D\ :\ D\to{\mathbb R}, \quad x\mapsto f(x)$ is continuous and similarly $f_S$ is continuous then $f_{D\cup S}$ need not be continuous on $D\cup S$. – Christian Blatter Mar 8 '12 at 15:54

The question could be understood in the following way: A function $f:\mathbb R\to\mathbb R$ is continuous on a set $A\subseteq\mathbb R$ iff it is continuous at each point $x\in A$.

If $f$ is continuous on $D$, that means that each $x\in D$ is a continuity point of $f$. Similarly, if $f$ is continuous on $C$, this means that each $x\in C$ is a continuity point. This obviously implies that $f$ is continuous at each point $x\in C\cup D$.

So if this is what OP has in mind when he speaks about a function continuous on a given subset of the domain, then the claim is true.

This is different from the answers which discuss continuity of the restrictions $f|_C$ and $f|_D$.

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What if I consider this example: f(x) = x if x is rational and f(x) = 0 if x is irrational. Then I will have that f is continuous only at 0. – max Mar 8 '12 at 13:45
As I said in my post, the whole thing depends on what you mean by by the term continuous on a set. It is true, that in your example both $f|_{\mathbb Q}$ and $f|_{\mathbb R\setminus\mathbb Q}$ are continuous. It is not true that this function is continuous at each point $x\in\mathbb Q$. It is not true that it is continuous at each point $x\in\mathbb R\setminus\mathbb Q$. – Martin Sleziak Mar 8 '12 at 14:14

Here’s a simple example of the situation that’s being described. Let $D=[0,1]$ and $S=[1,2]$, and define $$f(x)=\begin{cases}x,&\text{if }x\in[0,1]\\2-x,&\text{if }x\in[1,2]\;.\end{cases}$$ You can easily check that $f:[0,1]\to\Bbb{R}$ and $f:[1,2]\to\Bbb{R}$ are both continuous, as is $f:[0,2]\to\Bbb{R}$.

However, what happens if, for example, you take $D=\Bbb{Q}$, the set of rational numbers, and $S=\Bbb{R}\setminus\Bbb{Q}$? Can you define a function $f:\Bbb{R}\to\Bbb{R}$ such that $f$ is not continuous at any point, but $f\upharpoonright D$ and $f\upharpoonright S$ are both continuous?

For that matter, what if you take $D=[0,1)$ and $S=[1,2]$? Can you find a rather simple function that is continuous on $D$ and on $S$ but not on $[0,2]$?

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The result holds for closed sets.

Take $x_n\rightarrow a\in D\cup S$ then we have that part of this sequence pertains to $D$ and part pertain to $S$.

Let $(x_{n_j})\subset D$ then $f(x_{n_j})\rightarrow f(a)$ for the part that pertain to $D$, because $D$ is closed, for the part that pertains to $S$ we get $(x_{n_k})\subset S$ since $S$ is closed we have $f(x_{n_k})\rightarrow f(a)$ that implies $f(x_n)\rightarrow f(a)$, if one of the sequence is finite the result is achieved because for $n$ large enough $(x_{n})\subset D$ or $(x_{n})\subset S$ any way we will have $f(x_n)\rightarrow f(a)$ since the sequence was arbitrary we have $f$ is continuous.

The result is false for sets in general sets for a counter example take $f=1$ in $(0,1)$ and $f=0$ in $[1,2)$.

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Unfortunately, there is no guarantee that $(x_n)\subset S$ implies that $a\in S$ hence your proof breaks down. As it should... consider $D=\{x\mid x\leqslant0\}$ and $S=\{x\mid x\gt0\}$. – Did Mar 8 '12 at 10:46
Yeah the result is for both closed sets. It is false sets in general set for a counter example take $f=1$ in $(0,1)$ and $f=0$ \in $[1,2)$. – checkmath Mar 8 '12 at 11:06