Math Basic Analysis: Introduction to Real Analysis by Jiří Lebl

Math Basic Analysis: Introduction to Real Analysis by Jiří Lebl

Suppose f : [−1,0] → R and g: [0,1] → R are continuous and f(0) = g(0). Define h: [−1,1] → R by h(x) := f(x) if x ≤ 0 and h(x) := g(x) if x > 0. Show that h is continuous

Find an example of a bounded discontinuous function f : [0,1] → R that has neither an absolute minimum nor an absolute maximum

Let f : (0,1) → R be a continuous function such that lim x→0 f(x) = lim x→1 f(x) = 0. Show that f achieves either an absolute minimum or an absolute maximum on (0,1) (but perhaps not both).

Suppose g(x) is a polynomial of positive even degree d such that g(x) = x d +bd−1x d−1 +···+b1x+b0, for some real numbers b0,b1,...,bd−1. Suppose g(0) < 0. Show that g has at least two distinct real roots.

Suppose f : I → J is a continuous, bijective (one-to-one and onto) function for two intervals I and J. Show that f is strictly monotone