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A d.c. \(C^{1}\) function need not be difference of convex \(C^{1}\) functions. (English) Zbl 1121.26011

The author gives an example of a function \(f\:\mathbb R^2\to \mathbb R\), with the following properties: (i) \(f\in C^{1}(\mathbb R^2)\), (ii) \(f=f_1-f_2\), where both \(f_i\) are convex and continuous on \(\mathbb R^2\) (\(f\) is then called to be delta-convex), (iii) there are no \(f_i\) from (ii) which are differentiable at \(0\). Besides, he describes also an example of a delta-convex function \(f\:\mathbb R^2\to \mathbb R\) which is not strictly differentiable at \(0\), i.e. there exists no linear function \(L\) such that, for any \(\varepsilon >0\), \(| f(x)-f(y)-L(x-y)| <\varepsilon \| x-y\| \) for any \(x,y\) with \(\| x\| ,\| y\| <\delta \), where \(\delta =\delta (\varepsilon )>0\).

MSC:

26B25 Convexity of real functions of several variables, generalizations
26B05 Continuity and differentiation questions
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