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A) Give an example of a real valued function that has no fixed points

B) Give an example of a real valued function that has exactly two fixed points

C) Give an example of real valued function which is non linear and has infinitely many fixed points

D) Give an example of (non contractive) map which has a unique fixed point

for (a) I have found $f(x)=2$ when $x\leq1$ , $f(x)=x+\frac{1}{x}$ when $x\geq1.$

but I don't know for the rest of the questions.

Can any one help with this problem?

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    Any thoughts? I think, for A) something like $f(x)=x+1$ is simpler. – 2017-02-14
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    As a hint for the others: think about what sort of functions have, say, two zeroes. Try to adapt. – 2017-02-14
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    **Hint:** A fixed point has a very nice geometrical interpretation. Just draw it and find your functions! – 2017-02-14
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    Draw the line $y=x$ in the real plane and sketch a graph of a function $f$ that satisfies (A), (B), or (C). E.g. for (A), $f$ doesn't meet the line; for (B), $f$ meets the line exactly twice. For (D) try $f(x)=2x$. – 2017-02-14

1 Answers 1

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All functions in this answer have the domain $\mathbf R$.

A) $f(x)=x+1$. Obviously $f$ has no fixed point.

B) $f(x)=\begin{cases} 0 , &\quad x\leq 1 \\ 2x-1, &\quad x\in(1,10] \\ 29-x, &\quad x>10. \end{cases}$

$f$ has the two fixed points $x=2$ and $x=\frac{29}{2}$.

C) $f(x)=\begin{cases} 0 , &\quad x\leq \frac{1}{2} \\ 1, &\quad x\in \left(\frac{1}{2},\frac{3}{2}\right] \\ 2, &\quad x \in \left(\frac{3}{2},\frac{5}{2}\right] \\ 3, & \quad x\in \left(\frac{5}{2},\frac{7}{2}\right] \\ \dotsc \end{cases}$

$f$ has in every intervall $\left(\frac{2n+1}{2},\frac{2n+3}{2}\right]$ $(n\in \mathbf N_0$) the fixed point $x=2n+2$ and hence infinitely many fixed points.

D) $f(x)=\sin x$. The unique fixed point is $x=0$. $\sin$ is non contractiv by the mean value theorem since $|\cos x| \leq 1$ for all $x\in \mathbf R$.