I'm going through a set of older exams of a real analysis course to practise for the exam and I cannot prove the following:
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function with a finite number of local minima and maxima. Show that $f$ is then absolutely continuous.
I've been trying for hours but nothing has worked. I've tried appliying (obviously) the definition of continuity, then the same but near the local maxima and minima, went through several textbooks and found no useful theorem and the only thing I proved is that $f$ is of bounded variation (on a bounded and closed interval, of course) by splitting $f$ in several monotone & continuous functions. Thanks!