Provide an example of each of the following or explain why t

Provide an example of each of the following, or explain why the request is impossible A continuous function defined on an open interval with range equal to a closed interval. A continuous function defined on a closed interval with range equal to an open interval. A continuous function defined on an open interval with range equal to an unbounded closed set different from R. A continuous function defined on all of R with range equal to Q.

Solution

a) Consider the continuous function f(x) = sinx for x in R

Range is the closed interval [-1,1]but f is defined on the open interval (-infty, infty)

b) Let us take the continuous function f(x) = arcsin x, for x belonging to [-1,1] a closed interval.

But range is the open interval  (-infty, infty)

c) y =arctan x for -infty <x<infty is continuous function in the interval (-pi/2, pi/2)