Mathematics Homework Help

1. Let f and g be continuous functions on [a; b] such that f(a) g(a) and f(b) g(b).

Prove f(x0) = g(x0) for at least one x0 in [a; b].

5. Determine which of the following functions are uniformly continuous:

(a) f(x) = log x on (0; 1)

(b) f(x) = sin(cos(x)ejxj) on [ 1; 1]

(c) f(x) = ex on [0;1)

6. Give an example of each of the following, or state that such a request is impossible. For

any that are impossible, supply a short explanation for why this is the case. Assume

that all functions are dened on R.

(a) Functions f and g not dierentiable at zero but where fg is dierentiable at zero.

(b) A function f not dierentiable at zero and a function g dierentiable at zero where

fg is dierentiable at zero.

(c) A function f not dierentiable at zero and a function g dierentiable at zero where

f + g is dierentiable at zero.

(d) A function f dierentiable at zero but not dierentiable at any other point.

7. Let f(x) = x2 for x 0 and f(x) = 0 for x < 0.

(a) Show f is dierentiable at x = 0

(b) Calculate f0 on R

(c) Is f0 continuous on R? Is f0 dierentiable on R?