I'm trying to solve the below exercise.
Let be a continuous function. Prove that has at least one fixed point: an such that . Is the same true for discontinuous functions?
Here is my attempt.
Notice that if or , the theorem is proved. Suppose not. Define , which is a continuous function from to . Then since , , so . Since , . By the intermediate value theorem, there exists such that . So , so .
I am fairly sure that the result is not true for discontinuous functions, largely because I requiblack continuity of to invoke the intermediate value theorem. I am having trouble finding a counterexample, however. Do I define a function piece-wise, with jumps at or , to try to break the intermediate value theorem?