[Math] Showing that the function given by $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ and $f(0,0)=0$ is continuous but not differentiable

Question

Let
$$
f(x,y) =
\begin{cases}
\dfrac{xy}{\sqrt{x^2+y^2}} & \text{if $(x,y)\neq(0,0)$ } \\[2ex]
0 & \text{if $(x,y)=(0,0)$ } \\
\end{cases}
$$
Show that this function is continuous but not differentiable at $(0,0),$ although it has both partial derivatives existing there.

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I can show this function is continous and the partial derivatives exist. But how can I show that this function is not differentiable?

Is showing that the function is differentiable similar to showing that a derivative exists?

Answer 1

There are no directional derivatives in nearly all directions. Consider, in particular, along the line $y=x$. $f(x,y)$ is a constant times the absolute value function.

When a function of two variables is differentiable, then there is a tangent plane to the surface $z=f(x,y)$, and there are directional derivatives in all directions. This one doesn't have directional derivatives except in two directions, and there's no tangent plane to the surface $z=f(x,y)$.