# To what extent can one get rid of tangent lines and still have a continuous surface?

Does there exist in Euclidean 3-dimensional space R^3 a continuous 2-dimensional surface S specified by an equation of the form z-F(x,y) which satisfies the following conditions?

(1) F is continuous at each point (x,y) of a non-empty connected open subset of the x-y plane. (2) Given any arc c on S, c has no tangent at any of its points (or even no half-tangent at any of its points)

• What is a continuous surface? – Qiaochu Yuan Mar 23 '10 at 21:08

A typical realization of $W_1(s)+W_2(t)$, where $W_1$ and $W_2$ are standard independent Wiener processes, defines a function with requested properties.
UPD. The following example has no probability in it: take any continuous function $g:\mathbb{R}\to\mathbb{R}$ that has upper one-sided derivatives at each point equal to $+\infty$ and set $f(x,y)=g(x)+g(y)$.
• Your update is wrong. If $g$ is odd function then the surface $z=g(x)+g(y)$ contains a line. – Petya Mar 24 '10 at 11:23