MathDB

In whatever follows

f

denotes a differentiable function from

\mathbb{R}

\mathbb{R}

f \circ f

denotes the composition of

f(x)

<span class='latex-bold'>(a)</span>

f\circ f(x) = f(x) \forall x \in \mathbb{R}

then for all

x

f'(x) =

f'(f(x)) =

. Fill in the blank and justify.

<span class='latex-bold'>(b)</span>

Assume that the range of

f

is of the form

\left(-\infty , +\infty \right), [a, \infty ),(- \infty , b], [a, b]

. Show that if

f \circ f = f

, then the range of

f

\mathbb{R}

. (Hint: Consider a maximal element in the range of f)

<span class='latex-bold'>(c)</span>

g

satisfies

g \circ g \circ g = g

, then

g

is onto. Prove that

g

is either strictly increasing or strictly decreasing. Furthermore show that if

g

is strictly increasing, then

g

is unique.

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