### Home > APCALC > Chapter 2 > Lesson 2.1.2 > Problem2-27

2-27.

If the inverse of $f$ is a continuous function, why must the function be either strictly increasing or decreasing? Sketch an example to support your reasoning.

Strictly increasing means, as $x$ increases, $y$-values are always going up.
Strictly decreasing means, as $x$ increases, $y$-values are always going down.

If a function is NOT strictly increasing or decreasing, then either.
I. it oscillates between increasing and decreasing.
II. it is horizontal always or sometimes.

As you sketch, try to find a counter-example to this statement. In other words, try to sketch a function with a continuous inverse function that is NOT strictly increasing or strictly decreasing.

Note that $f(x)$ and its inverse must both be continuous AND both be functions.