I'm having trouble with the concept of continuous partial derivative and differentiability.
Let
$f(x,y) =\begin{cases} 1& \text{when } xy = 0\\0& \text{when } xy \neq 0\end{cases}$
then clearly, df/dx = 0 and df/dy = 0.
So both Partial Derivative are continuous and thus Differentiable everywhere?
using definition of Differentiability however, $\lim\limits_{(x,y) \to (0,0)} [f(x,y) - f(0,0) - 0 - 0]/[(x^2 + y^2)^{0.5}]$, the limit does not exist and therefore is not differentiable at (0,0).
So which is right?
and in the case, is $f(x,y)=\sin(x)\sin(x+y)\sin(x-y)$ differentiable at (0,0)
Thanks for all the help. Cheers!