Countinuity 98 Every rational number can be written in the f

Countinuity 98

Every rational number can be written in the from where n 0 and m and n are integers without any common divisors When 0, we take n 1. Consider the function f defined by f(s) Prove that f is continuous at every irrational point, and that f has a simple discontinuity at every rational point.

Solution

The irrational number cannot be written in the form of m/n, where m and n both are integers

Hence for every irrational number, the function will have the value of zero, since it cannot be represented in the form of m/n

f(\\sqrt 3-h) = f(\\sqrt 3) = f(\\sqrt 3 + h), where h is a very small quantity

The function will be discontinuous at every rational point, let us take an example of m=8 and n=5, then m.n will be equal to 1.6

f(1.6) = 1/5 =0.2

if we deviate from the rational number on both sides i.e. f(1.6-h) and f(1.6+h), the value will be equal to zero, hence the function will have a peak at 1.6 and then going down at remaining values

Hence the function will be discontinuous forevery rational point