micromass said:
Correct, but g necessarily takes the value 0 somewhere, right? That's all you need.JellyFox said:
A continuous function is a mathematical function that maintains its value as the input values change. This means that there are no abrupt changes or breaks in the graph of the function.
A fixed point of a function is a value that remains unchanged when the function is applied to it. In other words, if the input and output of a function are the same, then that value is a fixed point.
To prove that a continuous function has a fixed point, you can use the Intermediate Value Theorem. This theorem states that if a continuous function has different signs at the endpoints of an interval, then it must have at least one root (or fixed point) within that interval.
Yes, there are other methods to prove a continuous function has a fixed point. One method is the Brouwer Fixed Point Theorem, which states that any continuous function from a closed and bounded set to itself must have at least one fixed point.
No, a non-continuous function cannot have a fixed point. This is because a fixed point is a value that remains unchanged as the function is applied to it, and a non-continuous function will have abrupt changes or breaks in its graph, making it impossible for any input to remain unchanged.