No, it isn't. It's only differentiable at 0, and not throughout any neighbourhood.
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Yes. With w and z being any two numbers or variables: Take (w + z)2 = (w + z)(w + z) = ww + wz + zw + zz = w2 + 2wz + z2, which is the perfect square trinomial.
A=r mod z R= a relation which is reflexive symmetric but not transitive
All Euclid geometry can be translated to Analytic Geom. And of course, the opposite too. In fact, any geometry can be translated to Analytic Geom.
Analytic Geometry is useful when manipulating equations for planes and straight lines. You can get more information about Analytic Geometry at the Wikipedia. Once on the page, type "Analytic Geometry" into the search field at the top of the page and press enter to bring up the information.
An analytic function is a real valued function which is uniquely defined through its derivatives at one point.