## Absolute extrema and the Baire category theorem.(English)Zbl 1221.26006

In optimization problems it is very important to find points where a function $$f$$ continuous on a closed interval $$[ a , b ]$$ attains its maximum and minimum value. A method to solve such question is to find critical points (that is points $$x_0$$: $$f'(x_0) = 0$$ or $$f'(x_0)$$ does not exist), to compute the value of $$f$$ in these points and in the extremes $$a,b$$ and to compare these values. The authors present examples where such method is not useful (or practicable). The first example is a function with a countable set of critical points ($$f(x) = x \sin \frac{1}{x}$$ if $$0< x\leq 1$$, $$f(0) = 0)$$; the second is a function whose critical points are almost all the points of the interval (the Cantor function) and the third is a function with the set of critical points that equals its domain (the Weierstrass function).

### MSC:

 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

### Keywords:

optimization problems; special functions