This is exercise 7 from chapter 4 of Walter Rudin *Principles of Mathematical Analysis*, 3rd edition. (Page 99)

Define $f$ and $g$ on ${\bf R}^2$ by: $$f(x,y) = \cases {0,&if $(x,y)=(0,0)$\\
xy^2/(x^2+y^4) &otherwise}$$

$$g(x,y) = \cases {0,&if $(x,y)=(0,0)$\\
xy^2/(x^2+y^6) &otherwise}$$

Prove that:

1. $f$ is bounded on ${\bf R}^2$
2. $g$ is unbounded in every neighborhood of $(0,0)$
3. $f$ is not continuous at $(0,0)$
4. Nevertheless, the restrictions of both $f$ and $g$ to every straight line in ${\bf R}^2$ are continuous!

This is one of the few specific problems I remember from my university career, which ended some time ago. I remember it because I toiled over it for so long and when I finally found the answer it seemed so simple.