TITLE: Problem with Courant's 'Introduction to Calculus and analysis'
QUESTION [5 upvotes]: I am confused from 'Another example is furnished by the function $f(x) = x^2$.' I'm sure the previous pages and having read the book previously are not necessary as my problem is with the manipulations and not concepts.
I'm able to follow the manipulations but 
the confusion is as to how he arrived at the result 
$\delta$ $= -|x_o| + $($\epsilon$ $+|x_o|^2)^{1/2}$
Thank you for your time

REPLY [4 votes]: I'm able to follow the manipulations but the confusion is as to how he arrived at the result

Judging from the text, at this point you aren't expected to be able to come up with that formula, or any other suitable formula, for $\delta$ yourself. Its appearance here is just for exposition: simply to show that a formula does exist to make it work.
You should, however, be able to plug in the formula into the definition of continuity to check that it really does show that $x \mapsto x^2$ is continuous at $x_0$.
Also, and this very important: the formula is not a result. Working with the $\epsilon$-$\delta$ definition of limits is not a "solve for $\delta$" type of problem. When a limit exists, there will be a whole interval of values for $\delta$ that suffice, and as long as you find anything in that interval, you've succeeded.