# discontinuous and continuous

• Sep 3rd 2010, 06:56 PM
tsang
discontinuous and continuous
Give an example of a function $f: \mathbb R \rightarrow \mathbb R$ which is discontinuous at all points of $\mathbb Z$ and continuous at all points of $\mathbb R\\\setminus \mathbb Z$. Then prove the example has these properties.

• Sep 4th 2010, 04:02 AM
Ackbeet
What ideas have you had so far?
• Sep 4th 2010, 04:50 AM
Plato
Quote:

Originally Posted by tsang
Give an example of a function $f: \mathbb R \rightarrow \mathbb R$ which is discontinuous at all points of $\mathbb Z$ and continuous at all points of $\mathbb R\\\setminus \mathbb Z$. Then prove the example has these properties.

HINT: Think stair step.
• Sep 5th 2010, 12:39 AM
tsang
Quote:

Originally Posted by Plato
HINT: Think stair step.

Hi,thank you.But I still don't really understand what I can do.
Can I just say something like f(x)=1 for all $\mathbb R\\\setminus \mathbb Z$, and f(x)=0 for all $\mathbb Z$
• Sep 5th 2010, 03:54 AM
Plato
Quote:

Originally Posted by tsang
Hi,thank you.But I still don't really understand what I can do.

$f(x) = \left\lfloor x \right\rfloor$ was my suggestion.

But yes, f(x)=1 for all $\mathbb R\\\setminus \mathbb Z$, and f(x)=0 for all $\mathbb Z$ will work.