Answer to Question #192505 in Real Analysis for Nikhil

Question

Using the sequential definition of 
the continuity, prove that the funct-ion, f defined by
f(x)= { 3 , if x is irrational
      { -3 , if x is rational
 is discontinuous at each real numbers.

Answer 1

Ans:-

$f(x)=$ { $3$ , if x is irrational

{ $-3$ , if x is rational

$\Rightarrow$ This Function is discontinuous at each real number because According to sequential continuity

A function f : R→ R is said to be continuous at a point p ∈ R if whenever (a_n) is a real sequence converging to $p$ , the sequence $(f (a_n))$ converges to $f (p)$ .

$\Rightarrow lim_{h\to 0} f(x+h)= 3\\ \Rightarrow lim_{h\to0} f(x-h)=-3\\ \therefore lim_{h\to 0} f(x+h)\neq lim_{h\to0} f(x-h)$

Here at any real number if $f(x+h)$ become rational and and $f(x-h)$ become irrational according to sequential continuity. So . f(x) become discontinuous at each real number.