Can you tell us why you need three accounts?
RonL
Happy Thanksgiving to those of you in the States.
I have a problem:
Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
a<c<b.
Prove that f cannot be injective.
How would you work through this proof?
I am going to make this proof as if it was a maximum point the minimum point is anagolous.
Assume,
is an injective map.
Also that the function has a maximum .
Subdivide your interval into,
Since the function is continous on the full interval it is continous on these.
Without lose of generality assume . Pick any number such that
Then that number (by our conditions) also exists on,
By the intermediate value theorem there is a point such that on and on . Thus, , and if the function fails to be injective. The only way it can is when but that is not possible.
I hope you can understand it, mathematicians are by nature lazy.