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)].
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!