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In the paper "A literature survey of benchmark functions for global optimization problems" at this arxiv link https://arxiv.org/pdf/1308.4008.pdf, it describes the Cosine Mixture Function as "Discontinuous, Non-Differentiable, Separable, Scalable, Multimodal". The n-dimensional function is as follows: $$f(x)=-0.1\sum_{i=1}^n \cos{5\pi x_i}-\sum_{i=1}^n x_i^2$$ I'm not great at maths so it isn't immediately apparent where this function is discontinuous, and I can't see any areas where the function is undefined. Any help would be greatly appreciated as I'm pretty sure I am missing something.

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That categorization by the paper linked in the question is incorrect.

The function is indeed continuous and differentiable. In fact, it is continuously differentiable of all orders.

The "error" seems to lie in that linked paper, which references a previous paper, A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems, Montaz Ali, Charoenchai Khompatraporn, Zelda B. Zabinsky,  Journal of Global Optimization 31(4):635-672 · January 2005, which in turn references the original paper for that function, A deterministic algorithm for global optimization, Leo Breiman & Adele Cutler, Mathematical Programming volume 58, pages179–199(1993). However, neither of these previous papers make any statements about the continuity or differentiability of the function.

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  • $\begingroup$ Thanks for the references; I noticed another typo elsewhere in the document afterwards and wondered if this indeed was another one. $\endgroup$ Commented Apr 10, 2020 at 16:15

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