I am using Spivak calculus. Now Iam in the chapter limits. In pages 97-98, he has given the example of Thomaes function. What he intends to do is prove that the limit exists.(adsbygoogle = window.adsbygoogle || []).push({});

He goes on to define the thomae's function as

f(x)=1/q, if x is rational in interval 0<x<1

here x is of the form p/q where p and q are integers and q>0

f(x)=0, if x is irrational

He proves that the limit of f(x) as x tends to a(it is between 0 and 1) is 0

a can be either rational or irrational

He starts the arguement like this. He assumes a natural number n so large such that (1/n)<=€, ie..(1/n) is lesser than or equal to epsilon.

Since, L =0 ( The assertion he gives on value of this limit),

he writes that |f(x)-L| or |f(x)-0|<€,

He says that the numbers x which dont satisfy this inequality are...

1/2,

2/3

1/4,

3/4,

1/5,..

and he also includes

1/n....(n-1)/n..

My doubt is how?

Why dont these numbers satisy/ why is it false?

For example, consider x=1/n

so, f(x)=1/n

|1/n|<€. Is it not true? Isn't the inequality true?

Wont it lead to contradiction?

Where have I gone wrong?

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# Spivak Thomae's Function proof explanation

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