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) belonging to some 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)