# show a function of two variables is continuous

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• Feb 27th 2013, 11:12 PM
learning
Re: show a function of two variables is continuous
one question: in that case, is it ok to say $\sqrt{r^2} = r$? 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.
• Feb 27th 2013, 11:14 PM
Prove It
Re: show a function of two variables is continuous
Quote:

Originally Posted by learning
one question: in that case, is it ok to say $\sqrt{r^2} = r$? 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 \begin{align*} \sqrt{r^2} = |r| \end{align*}, not \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.
• Feb 27th 2013, 11:16 PM
learning
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.
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