Examples on boundedness, range and fixed points of functions

Let $\bar{B}^{\prime }(0,1)$ be the punctured closed unit ball of the space $\mathbb {R}^{n}$ (i.e., the ball with the center removed). Give an example of a continuous mapping of this set into itself having no fixed point.

Let $m[f]$ and $M[f]$ denote respectively the infimum and supremum of the function $f(x)$ on the interval $(a,b)$.

Prove that if $f_{1}(x)$ and $f_{2}(x)$ are functions defined on $(a,b)$, then

and

Give examples of functions $f_{1}(x)$ and $f_{2}(x)$ such that in the last two relations: (a) equality holds, (б) strict inequality holds.

Give an example of a function continuous on an interval:

Give an example of a discontinuous function, defined on a closed interval, whose range is a closed interval.

Give an example of a continuous function that takes the values 1 and 3, but does not take the value 2.

Give an example of a function, continuous on an interval, whose range is:

Give an example of a function:

Give an example of a function defined on a closed interval and unbounded on it.

For each of the following sets give an example of a continuous mapping of the set into itself that has no fixed point:

Give an example of a mapping of $[0,1]$ into itself that is not continuous and has no fixed points.