Absolute extrema and the Baire category theorem (Q2482932)

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).