A fixed point of a function f is a number c in its domain such that f(c) = c. (a) Sketch the graph of a continuous function wih domain [0,1] whose range also lies in [0,1]. Locate a fixed point f . (b) Try to draw the graph of a continuous function with domain [0,1] and range [0,1] that does not have a fixed point. What is the obstacle? (c) Use the Intermediate Value Theorem to prove that any continuous function with domain [0,1] and range [0,1] must have a fixed point.

Question

A fixed point of a function f is a number c in its domain such that f(c) = c.
(a) Sketch the graph of a continuous function wih domain [0,1] whose range also lies in
[0,1]. Locate a fixed point f .
(b) Try to draw the graph of a continuous function with domain [0,1] and range [0,1] that
does not have a fixed point. What is the obstacle?
(c) Use the Intermediate Value Theorem to prove that any continuous function with domain
[0,1] and range [0,1] must have a fixed point.

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Math
Calculus

Derivative