Question

2 11. Let f(c) be a continuous function defined on (a, b) whose range is (a, b). Show that there is a point on the graph of f that intersects the line y = r. That is, show that there exists a number cela, b] such that f(c) = c. Solution: If f(a) = a, then we can take c= a. If f(0) = b, then we can take c=b. Assume that f(a) +a and f(b) + b and consider g(1) = f(I) - I. Observe that since the range of f is [a, b] and f(a) +a, we have that f(a) > a. Hence, g(a) = f(a) - a>0. Similarly, observe that since the range of f is [a, b] and f(b) + b, we have that f(b)