Worksheet 8 The problem is due in the workshop on 1111 Wed 1

Worksheet 8 The problem is due in the workshop on 11/11 Wed. 1. Define a function f: [0, 1] ? > R as follows: (i) if x is irrational f(x) = 0, and (ii) if x is rational, f(x) = 1/q where x = p/q with p and q being natural numbers with no common divisors. (1) Show that f is not continuous at any rational number x0. (2) Show that f is continuous at any irrational number x0.

Solution

f(x) = 0 if x is irrational

     = 1/q where x =p/q a rational number in its least form

1) If f is continuous at p/q say then limit x tends to x0 = p/q left and right should be equal to 1/q

But when there is an irrational number near x0, then f(x0) = 0 hence left and right limit need not equal f(x0) = 1

Hence f is not continuous at any rational number

2) For an irrational number f(x0) = 0

Near the irrational numbers there may be again irrational numbers in the neighbourhood of x0 hence limit on both sides would equal 0.

Hence continuous at x=x0 is x0 is irrational