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Consider the function f : R R with f(x) = x if x is rational; 1 if x is irrational. (a) Show that f(x) is continuous at x = 1. (b) Show that f(x) is...

Consider the function f : R → R with

f(x) =    x   if x is rational;

             1  if x is irrational.


(a) Show that f(x) is continuous at x = 1.

(b) Show that f(x) is not continuous at x = 0.

Top Answer

(a) lim x->1 f(1) =1 and... View the full answer

5 comments
  • what about the discontinuity at 0
    • Giridhar
    • Dec 09, 2015 at 12:15am
  • (b) Given e > 0 (but less than 1) , choose d=e
    • ask_sumanto
    • Dec 09, 2015 at 12:34am
  • Then if |x|< d = e , we have |f(x) - f(0)| = |f(x)|=|x| < e if x is rational
    • ask_sumanto
    • Dec 09, 2015 at 12:35am
  • and |f(x) - f(0)| = |1- 0| =1 > e if x is irrational
    • ask_sumanto
    • Dec 09, 2015 at 12:37am
  • so, |x| < d does not imply |f(x) - f(0) | < e...so f(x) is not continuous at x=0
    • ask_sumanto
    • Dec 09, 2015 at 12:38am

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