Answer to Question #203201 in Real Analysis for Rajkumar

Question

Using the sequential definition of the continuity, prove that the function f , defined by:

f (x) = - 3, if is rational

f (x) = 3, if is irrational

is discontinuous at each real number.

Answer 1

f is continuous at a if and only if f(x_n)$\to$ f(a) for all sequences x_n , x_n$\to$ a.

Let a is rational, and sequence x_n$\to$ a. So, sequence x_n can contain as rational as irrational numbers. So, the terms of the sequence f(x_n) have only two values: 3 and -3. That is, the sequence f(x_n) will not have limit.

So, f is discontinuous at each real number.

Same thing is for the case when a is irrational.