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Continuous Functions

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Where is the function

f(x) = (q^2 - 1)/q^2 if x = p/q meaning x is a rational in reduced form and f(x) = 1 when x is not a rational

continuous in the interval (0,1)? Please also explain how you came up with the answer.

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Solution Summary

The continuity of a given function is investigated across an interval.

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(1) Let x0=p/q be a point in the interval (0,1) and x0 is an rational. By the definition of f(x), we know that
f(x0)=(q^2-1)/q^2. Obviously, f(x0)<1.
Since the irrational is dense in (0,1), we can choose a sequence y1,y2,...,yn,... such that yn->x0 and yi are irrational numbers for all i=1,2,....

By the definition of f(x), we have f(yi)=1 for i=1,2,...,
So, f(yn)->1 as n tends to infinity, which implies that the limit of f(x) (as x tends to x0=p/q) is not equal to f(x0)=(q^2-1)/q^2 if the limit of f(x) (as x->x0) exists.

So, by the ...

Solution provided by:
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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