Critical point but not an extremum or saddle point

Question

Let $f: R^2\to R$. Now, a critical point does not mean $f$ has a local (or global) extrema. Of course it could be a saddle point.

Does anyone have an example of a function $f: R^2\to R$ that has a critical point that is neither a saddle point nor a local (or global) extremum?

Answer 1

I don't think such function exists. In fact, one of many (equivalent) definitions of saddle point states

A point of a function or surface which is a stationary (critical) point but not an extremum.

http://mathworld.wolfram.com/SaddlePoint.html

Answer 2

I believe that such a function exist :

You may look at $f(x,y) = x^2 + y^3 - 3xy^2$

Then there are 2 stationary points : $(0,0)$ and $(1/6,1/3)$

You can check that $(1/6,1/3)$ is actually a saddle point. However, for $(0,0)$, we have $f(x,0) = x^2$ and $f(0,y) = y^3$

On the $x$-line it'd be a minimum. On the $y$-line it would be a saddle point. This point is neither an extremum, or a saddle-point.