Is this multivariable function differentiable

Question

I know that a function f(x,y) is differentiable at (a,b) Iff 1.fx exists at (a,b) 2.fy exists at (a,b) 3.fx is continuous at (a,b)

I checked in this case fx and fy exist at (0,0) Also fx is continuous at (0,0) But the answer in my book says that the function is not differentiable

I am confused now. Do I also have to check that fy is continuous at (0,0)?? I have never done that before but I never got this problem. Do I always have to check that fy is continuous at (0,0) or Checking fx is continuous sufficient. Is there any mistake in my work?

Answer 1

Note that $|f(x,y)| \le |y|$. This shows that $f$ is continuous at $(0,0)$.

Note that $f(t,t) = |t|$. This shows that $f$ is not differentiable at $(0,0)$.

It is easy to see that $f(t,0) = f(0,t) = 0$ from which is follows that the partials exist.