Proving the Intermediate Value Theorem and Range of 1:1 Continuous Functions

[busterkomo]

Homework Statement

a)Let f(x) be continuous on [0, 2], with f(0) = f(2). Show that f(x) = f(x+1) for some x ε [0, 1].
b)Let f(x) be 1:1 and continuous on the interval [a, b] with f(a) < f(b). Show that the range of f is the interval [f(a), f(b)].

Homework Equations

The Attempt at a Solution

I'm not really where to start for either of them. In a), I find it obvious that there exists an f(x) = f(x+1) for some x in that interval, but find it difficult to prove without any specific function. I find using the I.V.T. difficult in general without being applied to a specific function. Any help/hints appreciated. Thanks!

 

[Speedholic]

Hello fellow Waterloo student! I would love to help, but alas, I am having the same problems as you! :S

 

[mtayab1994]

Hello busterkomo try taking a function h such that:

h(x)=f(x)-f(x+1)

And h is obviously a continuous function as a difference of two continuous functions.
Now what can you do from that??