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

- 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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#### Other Answers

for full proof ask me question directly... View the full answer