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.

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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) = ixi1 for integers i 0.