Homework Statement
Suppose f:[a,b] \rightarrow [a,b] is continuous. Prove that there is at least one fixed point in [a,b] - that is, x such that f(x) = x.
Homework Equations
The Attempt at a Solution
I was going to try something with the IVT, but then I realized I wasn't sure what...
1.
Define f:R→R as follows:
f(x)= x- ⌊x⌋ if ⌊x⌋ is even.
f(x)= x- ⌊x+1⌋ if ⌊x⌋ is odd.
Determine those points where f has a limit and justify your conclusions (using δ and ε).
Homework Equations
3.
The attempt at a solution involved a graph of the situations of f(x)...