# Continuity of this function

• June 28th 2005, 04:09 AM
vms
Continuity of this function
f(x) is a function defined from Set of rational numbers Q to itself as

f(x)=x,

Is this function continuous at all rational numbers?
• June 29th 2005, 07:19 AM
hpe
Quote:

Originally Posted by vms
f(x) is a function defined from Set of rational numbers Q to itself as

f(x)=x,

Is this function continuous at all rational numbers?

It's continous from $\mathbB R$ to itself (just use $\epsilon = \delta$ in the definition of continuity), hence also continuous when restricted to any subset. So the answer is "yes, it's continuous".

On the other hand, in response to Ene Dene's post, if f(x) is defined as constant c for all real $x \notin \mathbB Q$, then f is continuous at x=c only .
• June 29th 2005, 10:56 AM
Ene Dene
Quote:

Originally Posted by hpe
On the other hand, in response to Ene Dene's post, if f(x) is defined as constant c for all real $x \notin \mathbB Q$, then f is continuous at x=c only .

I've deleted that post, after seeing your answer, becoase I thought I understood the question wrong. The question should have been more accurate.
Anyway, I'm satisfied now :).