Let $F:\mathbb{R}^2\to\mathbb{R}, F(x,y)=\sin |x^2+y|$.
a) Prove that $F(x,y)$ is not continuously differentiable at $(0,0)$.
b) Is $F$ differentiable at $(0,0)$?
I can do part b):
If $F$ is differentiable at $(0,0)$ then $\dfrac{\partial F}{\partial y}(0,0)$ must exist. However,
$$
\dfrac{\partial F}{\partial y}(0,0)=\lim\limits_{y\to 0}\dfrac{F(0,y)-F(0,0)}{y}=\lim\limits_{y\to 0}\dfrac{\sin|y|}{y}
$$ does not exist! So $F$ is not differentiable at $(0,0)$.\
I'm stuck at part a):
I have been trying to show that
$$
\dfrac{\partial F}{\partial x}(0,b)=\lim\limits_{x\to 0}\dfrac{\sin|x^2+b|-\sin|b|}{x}
$$ or that
$$
\dfrac{\partial F}{\partial y}(a,0)=\lim\limits_{y\to 0}\dfrac{\sin|a^2+y|-\sin|a^2|}{y}
$$ does not exist .
But I don't know how to proceed. Could someone help me or have other ways to deal with the problem? Thanks in advance!