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kamil9876:
Suppose that is a convex region bounded by a curve in . Prove that has a tangent line at every point except at a countable number of points.

kamil9876:
I think I've got an idea, just how to make it rigorous?

 Suppose some guy walks around the curve in an anti-clockwise fashion, starting at a point and doing a full circuit. Each time he moves his orientation changes by a certain angle. When walking along a section of the curve with only tangent points, the turn does not happen instantaenously, ie he must walk some distance before he changes his orientation. However at a non-tangent point, his orientation changes by a positive ammount instantaenously. Moreover his turns are always anticlockwise, never clockwise for otherwise the region is not convex. Therefore by having uncountably many non-tangent points, there are uncountably many instances where he makes a turn. His total turn is which is made up of smaller turns all of which are positive. But, informally speaking, the sum of uncountably many positive numbers* diverges, a contradiction.


*Of course, this doesn't make sense technically but what it means is this: consider an uncountable set of real numbers, there exists a countable subset whose sum diverges. This is a problem I have solved earlier. However our situation deals with a multi-set rather than a set since the angles can repeat (I think the old problem can probably be extended to this).

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