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.

Solution Preview

Solution.

(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 Summary

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

... n, for f (n) (0) where f is each of the functions (1) sin ... (Q.1) It is said that a function f : R ... differentiable at a point a ∈ R if: a) f is continuous at the ...

... X ~ Un(a,b), if its probability density function is constant ... For continuous independent random variables, their probability density functions are related ...

... X ~ Un(a,b), if its probability density function is constant ... For continuous independent random variables, their probability density functions are related ...

... µ , σ 2 ) . Then its characteristics is the probability density function. ( x−µ )2 − 1. f ( x) = 2σ 2 1 e 2π σ. 2. List some examples of continuous data ...

... Nominal scales for continuous attributes can be modeled but are ... of the arb() and mon() functions and the ... The function f() can involve any other transformations ...

... Since the first derivative of the solution function y (x )is continuous everywhere in ... Appendix: Integrating odd and even functions over symmetric intervals. ...

... use the receiving inspection function in lieu ... manufacturing to administrative functions and service ...continuous process improvement * customer focus * defect ...

... At 8% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place. 8)Suppose that the function represents the ...