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

(1) fF is continuous at each point (x,y) belonging to someof a non-empty connected open 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)

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)

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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Garabed Gulbenkian
Garabed Gulbenkian
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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) 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)