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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?

good question.

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 if S = [-3,-2] union [2,3]  then function is not continuous in the range (-2 to 2) because function does not exist in (-2,2)as S takes no value in this open interval
edited
need a strong reason.
Can you make $f(x)$ continuous for option C?