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Does there exist a convex function that is differentiable over an open convex set but not continuously differentiable there?

Any differentiable function defined on an interval is continuously differentiable due to the monotonicity and Darboux property of its derivative. Therefore, the function, if exists, has to reside in a $2$- or higher-dimensional space. In addition, it needs to be continuously differentiable along any straight line.

Any comments or criticism will be appreciated. Thanks.

Update: Indeed, there does not exist such a function. See [Corollary 25.5.1, Convex Analysis, Rockafellar 1970].

Nuno
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  • https://math.stackexchange.com/questions/3911841/differentiability-and-convex-function/3912463#3912463 – daw Nov 24 '20 at 15:21
  • Thanks. It seems that the example over there cannot be extended to a differentiable convex function on an open set that contains (0,0). What do you think? – Nuno Nov 24 '20 at 16:12

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