91Ó°ÊÓ

A function is called strictly increasing if whenever \(x_{1}\) and \(x_{2}\) are in its domain with \(x_{1}f\left(x_{2}\right)\).) Show that a function which is strictly increasing is one-to-one and that its inverse function is also strictly increasing. Prove also the corresponding result about a decreasing function. Give an example of a one-to-one function which is neither increasing nor decreasing.

Let \(f\) be the function given by $$ f(x)= \begin{cases}\frac{|x|^{2}+x|x|-2}{x-1} & x \neq 1 \\ 3 & x=1\end{cases} $$ Show that for each \(x_{0} \in \mathbb{R}\) the limit \(\lim _{x \rightarrow x_{0}} f(x)\) exists and that for \(x_{0} \neq 1\) the limit equals \(f\left(x_{0}\right)\) itself. Sketch the graph of \(f\). To what number would you change the ' 3 ' in the definition of \(f\) in order to get a function whose graph does not have a 'break"?