Extreme and Intermediate value theorem

[mikael27]

Homework Statement

Let f : [a; b] ! R be an arbitrary continuous function. Let S = {f(x)| a<= x<=b}. Show
that if S contains more than one element, then S is an interval of the form [c, d].

Hint: First apply the Extreme Value theorem, then the Intermediate Value theorem.

Homework Equations

The Attempt at a Solution

dont have any clue

 

[δοτ]

If we suppose S contains more than one points then are $a \le c_1 < c_2 \le b$ such that $f(c_1) \neq f(c_2)$. Now the EVT can be applied to say something about the relationship of these. Once that's established the MVT will show that it must be an interval.

This actually says something quite important about continuous mappings over real numbers.

 

[δοτ]

In response to your PM, the EVT can be applied to say that, in addition (without loss of generaltiy) $c_1$ and $c_2$ are the minimum and maximum on this interval, respectively. We can say this because continuous function must attain their maximum and minimum.

Now the IVT can be applied to show that $f$ also attains all values between $f(c_1)$ and $f(c_2)$. You can do the same for the intervals $[a,c_1]$ and $[c_2, b]$, then you have $f(c_2) = d > c = f(c_1)$ and so $f([a,b]) = [c,d]$.


It stil needs some details, but that's the gist of it.