Function where for which any change in xy-plane implies change in z

Question

I'm walking along the surface of some function f(x,y)=z. Now I come to a point (x0,y0,z0) from which I cannot move continuously to another point with the same z-coordinate, i.e., some (r,s,z0) where (r,s) -= (x,y). Does that imply that the function is not smooth? If there's a smooth function with the property, can you give me an example?

Two other ways of saying this, I think: [1] I'm walking along f in a neighborhood with no funny holes. Can I always move continuously from my coordinate (x0,y0,z0) to other points which have the same z, i.e., (r,s,z0). [2] Do smooth functions in 3 dimensions necessarily have continuous contour lines?

(I'm sorry if this has already been answered. I'm not sure how to search for it.)

Thank you!

Answer 1

Consider the example of $z=f(x,y)=|x|+|y|$

• This is gives a continuous surface
• One of the points you can reach is $(0,0,0)$
• If you are at that point, there is nowhere else on the surface, and so nowhere in a neighbourhood of that point, with $z=0$
• Contour lines are squares, though at that point the square is degenerate and has side $0$