Problem 1 : (10 points) Let f : R R be given by the "floor function" f(x) = [x] This means, that f(x) is the largest integer that is less than or...

Problem 1 : (10 points)

Let f : R → R be given by the "floor function" f(x) = [x]

This means, that f(x) is the largest integer that is less than or equal to x. So, for example, f(3.14) = 3, f(−2) = −2, f(−5.7) = −6.

1. Find, with justification, all x0 ∈ R at which f is continuous. 2. Find, with justification, all x0 ∈ R at which f is differentiable.

2

Problem 2 : (10 points)

1. Let a, b ∈ R. Show that f : R → R given by f (x) = ax + b is uniformly continuous on R.

2. Suppose f : R → R is uniformly continuous. Show that there are a, b ∈ R such that |f(x)| ≤ ax + b

for all x ∈ R.

3

Problem 3 : (10 points)

A set S of real numbers is called dense, if every (non-empty) open interval (a, b) contains at least one element of S.

Fix now a dense set S. Find, with justification, all continuous functions f : R → R such that f(x) = 0 for all x in S.

4

Problem 4 : (10 points)

Let n ≥ 1 be an integer and suppose a0,a1,··· ,an are real numbers such that a0+a1+···+an =0

Show that the equation

2 n+1

a0 +a1x+···+anxn =0

has at least one solution x0 with x0 ∈ (0, 1). Note:

You can assume without proof standard results on derivatives such as for example (xi)′ = ixi−1 for integers i ≥ 0.