I need a formal, detailed proof of the following. If f is continuous and increas

Question

I need a formal, detailed proof of the following.

If f is continuous and increasing on [a, b] or continuous and decreasing on [a, b], then for each y between f(a) and f(b) there is exactly one x [a, b] such that f(x) = y

This is the deffinition that we are given about increasing and decreasing functions

A function f is increasing on an interval I if x, y I and x < y imply f(x) < f(y). A function f is decreasing if x, y I and x < y imply f(x) > f(y). A function which is increasing or decreasing is said to be strictly monotonic.

Explanation / Answer

If f is continuous and increasing on [a, b] implies that f(a) < f(b)

or continuous and decreasing on [a, b] implies that f(a) > f(b).

  Suppose, for definiteness,

that f(a) < f(b) and f(a) < y < f(b).

(If f(a) > f(b) and f(b) < y < f(a), apply the same proof to f)

     Let g(t) = f(t) y. Then g(a) < 0 and g(b) > 0, so

  that g(x) = 0 for some a < x < b, meaning that f(x) = y.

(because we know that :

Suppose that f : [a, b] R is a continuous function on a closed, bounded interval. If f(a) < 0 and f(b) > 0, or f(a) > 0 and f(b) < 0, then there is a point a < c < b such that f(c) = 0.)

(Proved)

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