How to examine if multivariable functions are differentiable?

How to examine if functions:
$f(x,y)=|x+y|$
and
$g(x,y)=\sqrt{|xy|}$
are diffirentiable in points: $(0,0)$ for $f(x,y)$ and $(0,1)$ for $g(x,y)$

1 Answer

Normally, if you suspect a function to be differentiable, the easiest is to show that its partial derivatives exist and are continuous. This will imply differentiability (although it is not a necessary condition).

If you think a function is not differentiable in a given point, then you might be able to obtain a contradiction with the definition of differentiability by approaching the given point from various directions.

For example, in the first of your two cases, you may draw on your knowledge of what happens in the one-variable case and remember that $x \mapsto \lvert x \rvert$ is not differentiable in the point $0$ to come up with the guess that that something similar is probably also the case in two variables. More precisely, you can prove that $f$ is not differentiable in $(0,0)$ by approaching $0$ along the $x$-axis, letting $y= 0$ and obtaining two different results for the limits of the difference quotients (so that the limit does not exist).