0 votes 0 votes Let $S\subseteq \mathbb{R}$. Consider the statement “There exists a continuous function $f:S\rightarrow S$ such that $f(x) \neq x$ for all $x \in S.$ ” This statement is false if $S$ equals $[2,3]$ $(2,3]$ $[-3,-2] \cup [2,3]$ $(-\infty,\infty)$ Calculus isi2017-mma engineering-mathematics calculus continuity + – go_editor asked Sep 15, 2018 edited Nov 8, 2019 by go_editor go_editor 1.1k views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply Sukanya Das commented Mar 29, 2018 reply Follow Share I think answer will be A) $[2,3]$ 0 votes 0 votes Sukanya Das commented Mar 29, 2018 reply Follow Share What is the answer? 0 votes 0 votes Akash Papnai commented Oct 12, 2019 reply Follow Share I think option B is correct because consider the diagram given below as an example. Y=X+1 so, it's true for all options except for option b 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes http://mathworld.wolfram.com/FixedPointTheorem.html 2<=x<=3 value represented only rationals. so answer is (A) jjayantamahata answered Mar 26, 2018 edited Mar 29, 2018 by jjayantamahata jjayantamahata comment Share Follow See all 2 Comments See all 2 2 Comments reply srestha commented Mar 27, 2018 reply Follow Share why not C) ? 0 votes 0 votes Sukanya Das commented Mar 29, 2018 reply Follow Share Look at these References : http://mathworld.wolfram.com/FixedPointTheorem.html https://math.stackexchange.com/questions/2552345/is-there-exists-a-continuous-function-f-s-%E2%86%92-s-such-that-fx-%E2%89%A0-x-for-all-x-%E2%88%88-s?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes correct answer is option A, the complete explanation is provided in given link below , where the use of intermediate value theorem and fixed point property is used. (A) is false, i.e. [2,3]-> [2,3] has the Fixed Point Property. Consider a continuous function f:[2,3]→[2,3] and define g:[2,3]→R g(x)=f(x)−x It follows that g(2)≥0 and g(3)≤0. By the intermediate value theorem there is x0∈[2,3] such that g(x0)=0 . Thus f(x0)−x0=0 and so f(x0)=x0. https://math.stackexchange.com/questions/2552345/is-there-exists-a-continuous-function-f-s-%E2%86%92-s-such-that-fx-%E2%89%A0-x-for-all-x-%E2%88%88-s rishabhjain18 answered Jul 12, 2020 rishabhjain18 comment Share Follow See all 0 reply Please log in or register to add a comment.