Answer to Question #310432 in Real Analysis for Nikhil

Question

Is every onto strictly decreasing function invertible? Justify your answer.

Answer 1

Let as consider a strictly monotone surjective function $f:(a,b) \to (A,B).$

Without loss of generality let $f$ be increasing (otherwise consider $-f$ )

It is sufficient to show that the function is continuous, since each continuous strictly monotone function has an inverse.

Since $f$ is monotone, all of its points of discontinuity are of the first kind.

Let $x_*$ be such a point, $f(x_*-0),\, f(x_*+0)$ be the left and right limits at this point.

Then, since $f(x) < f(x_*-0)$ for $x < x_*$ and $f(x) > f(x_*+0)$ for $x > x_*$ it follows that

$(f(x_*-0), f(x_*+0)) \nsubseteq f((a,b))$ , which contradicts the surjectivity condition.

Hence, there are no points of discontinuity, or equivalently $f$ is continuous.