# Thread: Determine where f is continuous and where f is discontinuous

1. ## Determine where f is continuous and where f is discontinuous

I have never liked problems of the sort where rational/irrational numbers are involved.

For the function $\displaystyle f(x,y)=\left\{\begin{array}{cc}x^2+y^2,&x,y\mbox{ both rational }\\0, & \mbox{ otherwise, }\end{array}\right$
determine where $\displaystyle f$ is continuous and where $\displaystyle f$ is discontinuous. Does $\displaystyle f$ have any partial derivatives?

A note provided is as follows: Every neighborhood in $\displaystyle R^2$ always contains points with rational and points with irrational coordinates.

Mainly, the part I have difficulty with is the rational/irrational bit. I've never been able to work with those properly, probably because I missed something fundamental in my earlier years.

2. Originally Posted by Runty
I have never liked problems of the sort where rational/irrational numbers are involved.

For the function $\displaystyle f(x,y)=\left\{\begin{array}{cc}x^2+y^2,&x,y\mbox{ both rational }\\0, & \mbox{ otherwise, }\end{array}\right$
determine where $\displaystyle f$ is continuous and where $\displaystyle f$ is discontinuous. Does $\displaystyle f$ have any partial derivatives?

A note provided is as follows: Every neighborhood in $\displaystyle R^2$ always contains points with rational and points with irrational coordinates.

Mainly, the part I have difficulty with is the rational/irrational bit. I've never been able to work with those properly, probably because I missed something fundamental in my earlier years.
note that by the squeeze theorem that

$\displaystyle 0 < f(x,y)<x^2+y^2 \forall (x,y) \in \mathbb{R}^2$

Now just let $\displaystyle \epsilon=\delta$ and when

$\displaystyle |(x,y)-(0,0)|< \delta \iff x^2+y^2<\delta$

then $\displaystyle |f(x,y)-0|\le |x^2+y^2|=...$

Will this work for any other point than (0,0)?