- Let f: R to R be a continuous function. Show that if f(x+y) = f(x) + f(y) for all x, y in R, there is a real number c such that f(x) = cx.
- Let f be a bounded function on A. Show that
(a) sup_{x in A} kf(x) = k sup_{x in A} f(x)
and inf_{x in A} kf(x) = k inf_{x in A} f(x), if k greater than 0;
(b) sup_{x in A} kf(x) = k inf_{x in A} f(x)
and inf_{x in A} kf(x) = k sup_{x in A} f(x), if k less than 0.
- Let f: [0, 2 pi] to R be a continuous function such that f(0) = f(2 pi). Show that there is a real number c in [0, 2 pi] such that f(c) = f(c + pi).
- Show that any polynomial of odd degree with real coefficients has at least one real root.