Suppose that a function f is continuous on the
closed interval [a, b] and that a ≤ f(x) ≤ b for every x in [a, b]. Show that there must exist a number c in [a, b] such that f(c) = c.
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OCW Scholar - Single Variable Calculus
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|dw:1378088849100:dw|Draw a picture. We're looking at a domain of x values from a to b. We're told that the values of the function (in other words, the y-coordinates) are all in the range from a to b. Therefore the function must be entirely contained within a square space (it appears in my diagram as a rectangle) with a lower left corner (a,a) and upper right corner (b,b). I've drawn one possible function.
Now consider a line drawn from point (a,a) to (b,b) (not shown in my diagram). This line must always have at least one point in common with the line representing the function. Can you take it from here?