In a homework exercise, three functions are given, and it is asked which one of those is differentiable at the origin. The correct is answer is the third, which is indeed differentiable at the origin. My question, however, is why isn't the second one (shown below) differentiable?
$ f(x, y) = \begin{cases} \frac{xy^2}{x^2+y^2}& \Leftarrow (x, y) \ne (0, 0)\\ 0&\Leftarrow (x, y) = (0, 0) \end{cases}$
Both the partial derivatives at the origin exist and are continuous, so $f \in C^1$, and thus it should be differentiable at all points of its domain, in particular at the origin. To help understand the problem, I plotted the function, and indeed, it is not differentiable at the origin. So I ask: where did I made a mistake? Is my assessment about the partial derivatives wrong?
Thanks in advance for the help.