g(x) is a function which has a maximum and a minimum on , not (necessarily) continuous. f(x) is a continuous function. Does ∘ necessarily have a maximum on ?
I need either to find a counterexample, or to produce a proof.
I have difficulties with finding a discontinuous function, which has a maximum and a minimum on R, since I have to use a composite function later.
UPD: My (not very well-thought) suggestion would be f(x)=x^2, which is continuous, and g(x)=(x^3-1)/(x^2-1), which is discontinuous (at x=1, x=-1) but has both minimum and maximum (at (0,1) and (-2,-3) respectively).
∘ = ((x^3-1)/(x^2-1))^2, which has two minima, but no maxima.