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Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous and bounded function , then

$a$) $f$ has a fixed point.

$b$) $f$ cannot be increasing

$c$) $\lim_{x\rightarrow \infty} f(x)$ exists.

Now I think $a$) is correct. For $f$ being continuous and bounded there is a positive integer say $M$ such that $$|f(x)| < M$$ i.e. $$-M < f(x) < M$$ i.e. $$f(-M) > -M \text{ and } f(M) < M$$ i.e. if we take $$g(x)=f(x)-x$$ then it is continuous and $$g(-M) < 0$$ and $$g(M) > 0$$ and hence there is a point $x_0$ such that $$g(x_0)=0$$ i.e. $$f(x_0)=x_0$$

enter image description here

For option $c$) I drew this graph thinking it would be possible for the function $f$ to be increasing with $y=M$ being its asymptote but I am not sure since could not get the analytic definition of this . So if $c$) is wrong then it can be increasing then $f(x)$ being increasing and bounded will not the limit in $c$) exist? But may be not always . Need little help on proving $b$) and $c$) wrong .

Thanks..

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  • $\begingroup$ Your argument for the existence of a fixed point is correct except that I think you've got two inequalities going in the wrong direction. ${}\qquad{}$ $\endgroup$ – Michael Hardy Aug 29 '15 at 17:33
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b) is incorrect. Consider the function $f(x) = \tan^{-1}(x)$.

c) is incorrect. Consider the function $f(x) = \sin(x)$.

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