Re: show a function of two variables is continuous

one question: in that case, is it ok to say $\displaystyle \sqrt{r^2} = r$? wouldn't this lead to a positive and negative solution? and thus make two limits (i.e. the limit wouldn't exist)? i have the same problem when trying to solve this sticking with (x,y) values. what to do with the root.

Re: show a function of two variables is continuous

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Originally Posted by

**learning** one question: in that case, is it ok to say $\displaystyle \sqrt{r^2} = r$? wouldn't this lead to a positive and negative solution? and thus make two limits (i.e. the limit wouldn't exist)? i have the same problem when trying to solve this sticking with (x,y) values. what to do with the root.

I think I understand what you're saying. You can approach from any two paths and show you get different values. That would be enough to show the limit does not exist.

Edit: And yes, $\displaystyle \displaystyle \begin{align*} \sqrt{r^2} = |r| \end{align*}$, not $\displaystyle \displaystyle \begin{align*} r \end{align*}$, so it would be fine to show you get a positive or negative answer depending on which direction you choose to approach 0 from.

Re: show a function of two variables is continuous

but wouldn't the root automatically do that by itself? as you get two limits, a positive and a negative, even along the same direction? x-axis and y-axis are no good, but y=x? for example? leads to i believe (plus or minus) 1/2root2.