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  • $\begingroup$ What happens if $f$ is not increasing (or decreasing)? (I guess it can be still invertible if neither of those conditions is true) $\endgroup$ Commented Sep 25, 2015 at 15:40
  • $\begingroup$ In most cases that would be of interest, invertible implies strictly increasing or decreasing. More precisely, if $f$ is continuous (mapping real numbers to real numbers), and it's image is a connected subset, then it is either strictly increasing or strictly decreasing. You can convince yourself of this by drawing a few pictures. $\endgroup$ Commented Sep 25, 2015 at 15:46