$f:\mathbb{R}^2\rightarrow \mathbb{R}$ Defined by $f(x,y)= \frac{xy^2}{x^2+y^4}$ if $x\neq 0,y\in\mathbb{R}$ and $f(x,y)=0$ if $x=0,y\in\mathbb{R}$
Then
it is continuous but not differentiable at origin
differentiable at origin
has all first order partial derivative at origin.
does not have all first order derivatives at origin.
consider the limit $(x,y)\rightarrow (0,0)$ along the curve $y=m\sqrt{x}$ we get lim$(x,y)\rightarrow(0,0)\frac{x^2m^2}{x^2+m^4x^2}=\frac{m^2}{1+m^4}$ which is different for different values for $m$ hence $f$ is not continuous at origin, so 1 is false, and 2 is clearly false. I have checked that $f_x$ and $f_y$ exists at $(0,0)$ so only $3$ is correct and all others are false. could any one confirm me am I right? Thank you.