Question

If f:[a,b]→[a,b] is continuous, then f has a fixed point; that is, there is a c∈[a,b] such that f(c)=c

Answer 1

The student's question pertains to the fixed point theorem for continuous functions, which guarantees that any continuous function mapping an interval to itself has at least one point where f(c)=c.

Step-by-step explanation:

The concept referred to in the student's question is the fixed point theorem in the field of Mathematics, specifically within the topic of continuous functions.

According to this theorem, any continuous function f that maps an interval [a, b] into itself is guaranteed to have at least one fixed point within that interval; that is, there exists some value c in [a, b] such that f(c) = c.

This theorem applies to real-valued continuous functions and is a direct consequence of the Intermediate Value

Theorem, which ensures that if a continuous function takes on different values at two points, it must take on every value in between those points at some point within the domain.