Download The intermediate value property Definition 1. Let I an interval and f : I

The intermediate value property
Definition 1. Let I an interval and f : I → R a function. We say f has the intermediate
value property (IVP) if for any a, b ∈ I, a < b, and λ between f (a) and f (b), there is c ∈ (a, b)
such that f (c) = λ.
Problem 1. (Homework) Let f : I → R be a function. Then f has IVP if and only if
f (J) is an interval for any interval J ⊂ I.
Problem 2. Give an example of two functions f, g with IVP such that f + g doesn’t have
IVP.
Problem 3. Prove there are no continuous functions f : R → R such that f (x) is rational
if and only if f (x + 1) is irrational, for every x ∈ R.
Problem 4. If f : I → R is a monotonic function, then f has lateral limits in any point
x ∈ I.
Problem 5. Let f ; [0, 1] → [0, 1] be a function such that f (0) = 0, and f (1) = 1.
a) Prove that, if f has IVP then f is surjective.
b) Is there any surjective f which doesn’t have IVP?
Problem 6. If f : I → R is monotonic and has IVP, then f is continuous.
Problem 7. Prove that if f : I → R is monotonic, and f (I) is an interval then F is
continuous.
Problem 8. Let f, g : R → R be such that f is continuous, g is monotonic, and f (x) =
g(x) for any rational x. Prove that f = g.
Problem 9. a) Let f, g : R → R be periodic functions such that f + g has a limit at ∞.
Prove that f + g is constant.
b) If a cos ax + b cos bx ≥ 0 for any x ∈ R, then a = b = 0.
Problem 10. (Homework) Let f : R → R be a function with IVP, such that f ◦ f is
injective. Prove that f ◦f is increasing.