## 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

### Keywords:

diferentiability; delta-convex functions
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