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True or False? Counterexample or Proof?
a) Every decreasing function from R to R is surjective
b) Every nondecreasing function from R to R is injective
c) Every injective function from R to R is monotone
d) Every surjective function from R to R is unbounded
e) Every unbounded function from R to R is surjective
f) f(x) = ax + b is both surjective and injective
My thoughts (and HELP on areas of WHY they are or are not).
a. true, HELP.
b. false, f(x)=x^2 for example.
c. no idea, HELP.
d. true, HELP
e. true, HELP
f) true, HELP.
Thanks so much.
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Do functions have to be continuous? If no, then...
a) Every decreasing function from R to R is surjective
False. f(x) = -x for all x <= 0, f(x) = -x - 1 for all x > 0.
b) Correct.
c) False. f(x) = -x for all x <= 0, f(x) = x - 1 for all 0 < x < 1, f(x) = -x - 2 for all x >= 1.
d) Every surjective function from R to R is unbounded
True. Let f(x) be a surjective function that is bounded by M. By the Archimedean Principle, there exists a b in R such that y > M. Thus, there exists an a in R such that f(a) = b > M. Contradiction.
e) Every unbounded function from R to R is surjective
False. Just cause a function goes to infinity does not mean it goes to negative infinity.
f) f(x) = ax + b is both surjective and injective
True. Just show 1-1 and onto, quite simple to do.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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I think (b) is false:
Nondecreasing is not the same as increasing.
And (c) would be true if f is continuous.
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