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:
$$f(x,y)=\begin{cases} \sin(y-x) & \text{for} & y>|x| \\ \\ 0 & \text{for} & y=|x| \\ \\ \frac{x-y}{\sqrt{x^2 + y^2}} & \text{for} & y<|x| \end{cases}$$
- Study $f$ with respect to continuity on its domain.
- 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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$\begingroup$Usually Schaum's outlines are a good source for lots of problems with solutions, in this case
- Elliott Mendelson: Schaum's 3,000 Solved Problems in Calculus
and
- Robert Wrede and Murray Spiegel: Schaum's Outline of Advanced Calculus
come to mind. I don't know if that's hard enough, however :-)
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