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I'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not.

For example, a typical problem goes like this:

Let $f:\mathbb{R^2}\longrightarrow\mathbb{R}$ defined by:

$f(x,y)=\begin{cases} \sin(y-x) & \text{for} & y>|x| \\ \\ 0 & \text{for} & y=|x| \\ \\ \frac{x-y}{\sqrt{x^2 + y^2}} & \text{for} & y<|x| \end{cases}$

  1. Study $f$ with respect to continuity on its domain.
  2. Study $f$ with respect to differentiability on its domain.

I think I know how to solve this, but I have no way to verify my answer and I might be unaware of some subtleties. Moreover, I did some browsing, but I was unable to find examples containing functions defined with branches such as this one. As you probably know, branches are precisely what make this a non-trivial problem (at least for me!).

So, I came here to ask for recommendations on books or online resources with solutions (don't need all the details, just the results) to problems like this one.

Thanks!

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    For this particular problem I suggest introducing new euclidean coordinates $(u,v)$: Put $x:={1\over\sqrt{2}}(u-v)\ ,\quad y:={1\over\sqrt{2}}(u+v)\ .$2012-01-26

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Usually Schaum's outlines are a good source for lots of problems with solutions, in this case

and

come to mind. I don't know if that's hard enough, however :-)

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    One can always raise the difficulty by trying to answer the problems while playing a flute and a sitar simultaneously.2018-03-19