In this exercise, we provethat a continuous function f mapping the unit interval [0, 1] to itself must have a fixed point—a point c satisfying f(c) = c (so f doesn't move or change the input). We start by discussing examples of fixed points and develop an intuition for how such points might occur. By sketching different continuous functions and analyzing their behavior, we understand how to graphically represent a fixed point.
Then we use a well-known result from calculus (spoiler: the Intermediate Value Theorem) to formalize this idea, applying it in a creative way to a new function that we create.
At the end, I mention how this principle connects to real-world situations, such as laying a rope back on itself (with MATLAB demos) or creating a "You Are Here" point on a map.
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