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Math Help - monotonic functions

  1. #1
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    monotonic functions

    Let f be defined on an interval I and suppose that f is one to one on I.

    (a) Give an example to show that f may not be monotone on I.
    (b) Give an example to show that f may not be monotone on any subinterval of I.
    (c) Suppose that f is continuous on I. Prove that f is monotone on I.
    (d) Suppose that f has the intermediate value property on I. Prove that f is monotone on I.
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  2. #2
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    Quote Originally Posted by friday616 View Post
    Let f be defined on an interval I and suppose that f is one to one on I.

    (a) Give an example to show that f may not be monotone on I.
    (b) Give an example to show that f may not be monotone on any subinterval of I.
    (c) Suppose that f is continuous on I. Prove that f is monotone on I.
    (d) Suppose that f has the intermediate value property on I. Prove that f is monotone on I.
    For (a) and (b) let f(x)= x if x \in I \cap \mathbb{Q} and f(x)=-x if x \in I \cap (\mathbb{R} - \mathbb{Q}). For (d) suppose x<y and f(x)<f(y) (other possible cases are similar) and suppose there is a z \in (x,y) such that f(z)<f(x)<f(y) then by the intermediate value property there is a c \in (z,y) such that f(c)=f(x) so f is not one-one. Similarly there is no z \in (x,y) such that f(z)>f(y) and so for all z \in (x,y) f(x)<f(z)<f(y). Now pick any a,b \in I a<b and show that f(a)<f(b) (Consider the cases where (a,b) and (x,y) are disjoint and when they're not separately). Since a continous function satisfies the ivp, you're done.
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