MATH 295 PROBLEMS 8

IGOR KRIZ

1. Suppose that f : [0, 1] → [0, 1] is a continuous function. Prove that there is an x ∈ [0, 1] such that f(x) = x. [Hint: use the equation solving theorem.]

2. How many continuous functions f : ℝ → ℝ are there which satisfy (f(x))^{2} = x^{2} for all x?

3. Prove that the function f(x) = \sqrt{x} is continuous at x ≥ 0.

4. Define ( \begin{matrix}n \\ k \end{matrix} ) = \frac{n!}{k!(n−k)!}. [Recall that n! = 1.2. ... .n.]

(a) Prove that ( \begin{matrix}n + 1 \\ k + 1 \end{matrix} ) = ( \begin{matrix}n \\ k + 1 \end{matrix} ) + ( \begin{matrix}n \\ k \end{matrix} ).

(b) Prove the “binomial theorem”: If a and b are any numbers an n is a natural number, then

(a + b)^{n} = a^{n} + ( \begin{matrix}n \\ 1 \end{matrix} ) a^{n−1}b + ( \begin{matrix}n \\ 2 \end{matrix} ) a^{n−2}b^{2} + ... + ( \begin{matrix}n \\ n − 1 \end{matrix} ) ab^{n−1} + b^{n}.

5. Prove or give a counterexample to each of the following assertions:

f ∘ (g + h) = f ∘ g + f ∘ h. (1) (g + h) ∘ f = g ∘ f + h ∘ f. (2)

\frac{1}{f ∘ g} = \frac{1}{f} ∘ g. (3)

\frac{1}{f ∘ g} = f ∘ (\frac{1}{g}) . (4)

[Here ∘ denotes composition of functions, + denotes addition of functions.]