Can a Function Be Discontinuous Only at Irrationals?

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  • Thread starter Euge
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    2016
In summary, A discontinuous on irrationals real-valued function is a mathematical function that is not continuous at any irrational number in its domain, meaning it has gaps or jumps in its graph at these points. You can identify if a function is discontinuous on irrationals by looking for vertical asymptotes or jump discontinuities at irrational numbers, or by evaluating the limit at these points. These functions are significant in mathematics as they show the limitations of continuous functions and can be used to construct more complex functions. They can also be used in real-world applications to model problems involving irrational numbers and to study the behavior of real numbers.
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Euge
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MHB
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Here is this week's POTW:

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Does there exist a real-valued function on $\Bbb R$ that is discontinuous only on the irrationals?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
No one answered this week's problem. You can read my solution below.
No. If so an function $f$ existed, then its oscillation $\omega_f$ would be identically zero on $\Bbb Q$. The rationals can then be written as a countable intersection of open sets $A_n := \{x : \omega_f(x) < 1/n\}$. This implies $\Bbb Q$ is a G$_{\delta}$ set, in $\Bbb R$, contradicting the Baire category theorem.
 

FAQ: Can a Function Be Discontinuous Only at Irrationals?

What is a "Discontinuous on Irrationals" real-valued function?

A discontinuous on irrationals real-valued function is a mathematical function that is not continuous at any irrational number in its domain. This means that there are gaps or jumps in the graph of the function at these points.

How can you identify if a function is discontinuous on irrationals?

A function is discontinuous on irrationals if it has a vertical asymptote or a jump discontinuity at any irrational number in its domain. This can also be identified by evaluating the limit of the function at these points and seeing if it exists or not.

What is the significance of a discontinuous on irrationals real-valued function?

Discontinuous on irrationals functions are important in mathematics because they demonstrate the limitations of continuous functions. They also play a role in understanding the behavior of functions near irrational numbers and can be used to construct more complex functions.

Can a discontinuous on irrationals function be continuous at rational numbers?

Yes, a discontinuous on irrationals function can be continuous at rational numbers. This means that it can have a smooth graph without any gaps or jumps at these points, but still have discontinuities at irrational numbers.

How can discontinuous on irrationals functions be used in real-world applications?

Discontinuous on irrationals functions can be used to model real-world problems that involve irrational numbers. For example, they can be used in physics to model the motion of a pendulum or in economics to model the growth of a population. These functions can also be used to study the behavior of real numbers and their relationships with irrational numbers.

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