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the statement comes from Natanson's real analysis.Firstly,he shows a theorem which says that if $f(x)$ is a bounded low-convex function in $[a,b]$,then $f(x)$ is continuous in $(a,b)$,then he appends to say that "the condition bounded is very important,there exist a finite,nowhere continuous convex function that is not bounded in any interval".

I don't know how to construct such funtion,what exists in my mind so far is the fact that convex function is always continuous(at least in finite dimension).But from what he said it seems exist such function defined on $\mathbb R$.Is it a contradiction?How to construct such function

math
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  • Perhaps the author defines convex as “midpoint convex”? Otherwise the statement is wrong and you are right. A convex function on an open real interval is continuous. – Martin R Aug 11 '21 at 08:43
  • @MartinR. You are right,several pages before,he does define it as midpoint,does it mean if we define convex function in this way,then there exists such fuction? – math Aug 11 '21 at 08:51
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    Yes, see https://math.stackexchange.com/q/71019/42969 and https://math.stackexchange.com/q/3637220/42969. – Martin R Aug 11 '21 at 08:51

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