## 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)

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What is a continuous surface? – Qiaochu Yuan Mar 23 2010 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.

Of course, there are simpler examples.

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 $f(x,y)=g(x)+g(y)$.

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Your update is wrong. If $g$ is odd function then the surface $z=g(x)+g(y)$ contains a line. – Petya Mar 24 2010 at 11:23
Let U be the non-empty open connected subset of the Cartesian x-y plane at all points of which F(x,y) is defined and continuous. What makes this problem difficult is that once having specified F(x,y), one then has to prove that no arc in R^3 with an equation of the form "x=g(t),y=h(t),z=F(g(t),h(t))-where t lies in the closure of (0,1)" has a tangent at any of its points, whenever x=g(t),y=h(t)is the equation of an arc lying in U. – Garabed Gulbenkian Apr 15 2010 at 20:27