Isolated continuity point

In summary, the conversation discussed the possibility of a real function being continuous at a certain point within its domain but discontinuous in a neighborhood of that point. An example of such a function was given, and the concept of an isolated continuity point was also brought up. The conversation then shifted to finding an example of a function that is differentiable only at one point, and a discussion on the construction of such functions using rational and irrational numbers took place. The question of whether it is possible to find such functions without using dense subsets of the real numbers was also raised.
  • #1
glance
8
0
Hi
The question is the following: is it possible for a (say) real function to be continuous at a certain point internal to its domain, and be discontinuous in some neighborhood of that point?
I am not talking about a function defined at a single point or things like that, but of a function defined on the entire [itex]\mathbb{R}[/itex] (or some interval in it, whatever).

Now, i have also came up with an answer: a function [itex]f[/itex] defined as [itex]f(0)=0[/itex], [itex]f(x)=x[/itex] for every rational [itex]x[/itex], and [itex]f(x) = 2x[/itex] for every irrational [itex]x[/itex]. Such a function would be (seems to me) continuous at [itex]x=0[/itex] and discontinuous for any other [itex]x[/itex]. I am not completely certain of this, though, and for that reason i would like some feedback on this.

I am also asking this question because strangely enough I have never heard of the concept of an isolated continuity point, while for example the "opposite" (that of an isolated singularity) is quite common, and I would like to know if it's just me or if it is just a "useless" pathological concept.

Thank you in advance.
Bye
 
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  • #2
glance said:
Is it possible for a (say) real function to be continuous at a certain point internal to its domain, and be discontinuous in some neighborhood of that point?

Yes.

In fact, there is a function defined on all of ##\mathbb{R}## which is continuous at a single point.
 
  • #3
glance said:
Now, i have also came up with an answer: a function [itex]f[/itex] defined as [itex]f(0)=0[/itex], [itex]f(x)=x[/itex] for every rational [itex]x[/itex], and [itex]f(x) = 2x[/itex] for every irrational [itex]x[/itex]. Such a function would be (seems to me) continuous at [itex]x=0[/itex] and discontinuous for any other [itex]x[/itex].

Yes, that is a correct example.

Now, can you come up with an example of a function defined on entire ##\mathbb{R}## that is differentiable only in one point? :tongue:
 
  • #4
glance said:
Hi
The question is the following: is it possible for a (say) real function to be continuous at a certain point internal to its domain, and be discontinuous in some neighborhood of that point?
I am not talking about a function defined at a single point or things like that, but of a function defined on the entire [itex]\mathbb{R}[/itex] (or some interval in it, whatever).

If [itex]f : \mathbb{R} \to \mathbb{R}[/itex] and [itex]g : \mathbb{R} \to \mathbb{R}[/itex] are continuous such that there exists a unique [itex]a \in \mathbb{R}[/itex] such that [itex]f(a) = g(a)[/itex], then the function [tex]
h : \mathbb{R} \to \mathbb{R} : x \mapsto \begin{cases} f(x) & x \in \mathbb{Q} \\
g(x) & x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} [/tex] is discontinuous on [itex]\mathbb{R} \setminus \{a\}[/itex] and continuous at [itex]a[/itex].
 
  • #5
pasmith said:
If [itex]f : \mathbb{R} \to \mathbb{R}[/itex] and [itex]g : \mathbb{R} \to \mathbb{R}[/itex] are continuous such that there exists a unique [itex]a \in \mathbb{R}[/itex] such that [itex]f(a) = g(a)[/itex], then the function [tex]
h : \mathbb{R} \to \mathbb{R} : x \mapsto \begin{cases} f(x) & x \in \mathbb{Q} \\
g(x) & x \in \mathbb{R} \setminus \mathbb{Q} \end{cases} [/tex] is discontinuous on [itex]\mathbb{R} \setminus \{a\}[/itex] and continuous at [itex]a[/itex].
That is a very interesting example, thank you.

micromass said:
Now, can you come up with an example of a function defined on entire ##\mathbb{R}## that is differentiable only in one point? :tongue:
That seems to be tricky! I did some research and stumbled upon this discussion of that matter, in which that question is very well explained.

Now however I wonder if it is accidental that in all of these examples the functions are constructed using rational and irrational numbers. I think that the important point is to have one subset which is dense in the other. Is there some example of functions having this kind of "pathologies" NOT using rational/irrational numbers in the definition?
Even better, is it possible to find a function of this kind NOT using at all dense subsets of the real numbers in the definition?

Thanks
 

1. What is an isolated continuity point?

An isolated continuity point is a point on a function where the function is both continuous and discontinuous. This means that the function is continuous at that point, but is not continuous at any other point in its domain.

2. How can you identify an isolated continuity point?

An isolated continuity point can be identified by checking for a break in the function's graph at a specific point. This break indicates that the function is discontinuous at that point, but still continuous at all other points.

3. Can an isolated continuity point exist on a continuous function?

Yes, an isolated continuity point can exist on a continuous function. This is because a continuous function can have a break in its graph at a specific point, while still being continuous at all other points.

4. What is the significance of an isolated continuity point?

An isolated continuity point is significant because it helps us understand the behavior of a function. It indicates that the function is continuous at some points and discontinuous at others, which can affect the function's overall behavior and properties.

5. How can isolated continuity points be useful in real-life applications?

In real-life applications, isolated continuity points can help us understand the behavior of natural phenomena, such as weather patterns, population growth, or stock market trends. They can also be used in engineering and physics to analyze and predict the behavior of systems and processes.

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