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Let S ⊆ R. 

Consider the statement:

"There exists a continuous function f:S -> S such that f(x) != x for all x belongs to S."

This statement is false if S equals

A) [2,3]          B) (2,3]          c) [-3,-2] union [2,3]         D) (-infinity to +infinity)

I think the answer should be D because in -infinity to +infinity, f(x) will definitely be equal to x at some point but we want this to not happen as said in the question. Am I correct?

asked in Set Theory & Algebra by Active (1.5k points)   | 53 views

1 Answer

+1 vote
Best answer
good question.

yes u r correct . because we do not know the equation of function hence we can not predict at what point it is discontinuous. but we can definitely say that open interval (-infinity to +infinity) will include all possible number ,hence whatever will the function be , at some point it may violate the given condition . therefore option D is correct one
answered by Active (1.8k points)  
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need a strong reason.


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