- #1
Portuga
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Consider a continuous function [itex]f[/itex] in [itex][a,b][/itex] and [itex]f(a) < f(b)[/itex]. Suppose that [itex]\forall s \neq t[/itex] in [itex][a,b][/itex], [itex]f(s) \neq f(t)[/itex]. Proof that [itex]f[/itex] is strictly increasing function in [itex][a,b][/itex].
I.V.T: If [itex]f[/itex] is continuous in [itex][a,b][/itex] and [itex]\gamma[/itex] is a real in [itex][f(a),f(b)][/itex], then there'll be at least one [itex]c[/itex] in [itex][a,b][/itex] such that [itex]f(c) = \gamma[/itex].
This exercise is very strange to me. Besides I can apply the I.V.T to show that for any sub interval in [a,b] there will be an intermediate value in [itex]f(a), f(b)[/itex], I can easily draw and counter example of what it pretends:
https://www.dropbox.com/s/dtj28xo4ilaai4z/pf.eps?dl=0
I am missing something important?
Thanks in advance!
Homework Equations
I.V.T: If [itex]f[/itex] is continuous in [itex][a,b][/itex] and [itex]\gamma[/itex] is a real in [itex][f(a),f(b)][/itex], then there'll be at least one [itex]c[/itex] in [itex][a,b][/itex] such that [itex]f(c) = \gamma[/itex].
The Attempt at a Solution
This exercise is very strange to me. Besides I can apply the I.V.T to show that for any sub interval in [a,b] there will be an intermediate value in [itex]f(a), f(b)[/itex], I can easily draw and counter example of what it pretends:
https://www.dropbox.com/s/dtj28xo4ilaai4z/pf.eps?dl=0
I am missing something important?
Thanks in advance!