Periodicity of function: f(x)={1,when x is rational; 0,when x is irrational}
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Steven G
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Can you give me one non empty interval where f(x) is constant?
Could you elaborate
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There are infinitely many irrational numbers between two given rational numbers.
What is the definition of a periodic function?
Does it have to be continuous?
Does it have to be continuously differentiable?
To the last two questions- no and no. I want you to look up the definition and then decide. If someone already has knowledge on this subject that would be preferred.
HallsofIvy
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Since you were the one who started these questions wouldn't it be better for you to look up the definitions yourself?
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Steven G
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Subhotosh knows the answers to the question but wants to make sure that you know them. Subhotosh has quite a bit knowledge on this subject. It appears to me that you just want the answer. If that is the case, then you came to the wrong website as you only receive help here as the title of the website suggests. Now before you leave another reply you need to answer Subhotosh's questions in detail before you will get any help. And also maybe YOU should look up the definition of a periodic function and think why Subhotosh and I asked you those questions.
Steven G
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Which do you not know, what a non empty interval is or what a constant function is? Maybe you should look these terms up.
Look I know this stuff. I know this function is periodic although the period is not defined. I just found it a little bizarre for such a function to be periodic. So I just wanted an expert to analyse it and sort of state what they think. Is that alright?
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Great! Then you can reply with your answers to the posted questions.
On what basis have you determined the function is periodic, especially as it has no period? :shock:
Please be complete. Thank you!
Dr.Peterson
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Maybe it would be good to start fresh, with your real question. Here is what I think you really wanted to ask:
f(x)={1,when x is rational
0,when x is irrational}
I know this function is periodic although the period is not defined. I found it a little bizarre for such a function to be periodic. What do you think about it?
0,when x is irrational}
I know this function is periodic although the period is not defined. I found it a little bizarre for such a function to be periodic. What do you think about it?
Yes, it is indeed a bizarre example; this function, and others like it (sometimes called pathological functions), are given specifically for that purpose, to stretch our minds and help us see where we may have oversimplified images of concepts like periodicity.
So to start a conversation about it, I'd like to lead you through some thinking, rather than just tell you my off-the-cuff answer, so we can deal with your specific issues. That can begin with you showing how you proved that it is periodic, so we can then look at that together.
Note that the function actually has many periods; when you say the period is not defined, you probably mean it has no fundamental period. This may be a point on which it will be helpful for you to not only show your work, but also state the definitions you have been given, as this is one where books sometimes differ (though I would think that a textbook that gives this exercise would have given very careful definitions!). For example, Wikipedia distinguishes between "a period" and "the fundamental period"; MathWords does not, so that its definition is deficient. It sounds like you used the latter approach, which is not appropriate for this exercise.
After we get past the basics, we can discuss what you see as bizarre, and then how that can help us correct misunderstandings of concepts like periodicity.
HallsofIvy
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It is incorrect to say that this function does not have a period. f(x) is "periodic with period p" if and only if f(x+ p)= f(x) for all x. Since any rational number plus a rational number is a rational number is rational, and any irrational number times plus a rational number is an irrational number, this function has every rational number as period. If the set of all periods of a function is a smallest positive number, we call that the fundamental period. This function has no fundamental period.