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I am stuck at the following problem, I got in an old question paper (real analysis).

Let $k>0$ be a natural number and Let $f$ be a continuous function on real line such that $f$ takes any value at most $k$ times. Show that $f$ is differentiable almost everywhere.

My hunch is that we can divide the real line (except a few exceptional points) into intervals such that on each of those intervals $f$ is one-to-one and therefore monotonic and hence differentiable almost everywhere. But, I am not able to make this idea concrete. Any help would be appreciated.

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It suffices to show that under the given condition, $f$ is locally of bounded variation, because then $f$ can be written as a difference of two continuous non-decreasing functions, which are differentiable almost everywhere by the Lebesgue's theorem for the differentiability of monotone functions. So the following proposition gives the result.

Proposition: If $f$ is continuous on $[a,b]$ and takes each value at most $k$ times, then it holds $$ V_a^b(f) \le k (M-m) $$ where $V_a^b(f)$ is the total variation of $f$ on $[a,b]$, and $\displaystyle M = \max_{x\in [a,b]}f(x)$ and $\displaystyle m = \min_{x\in [a,b]} f(x)$.

Proof: Take any partition $\Pi = \{a=x_0<x_1<\cdots<x_{n-1}<x_n = b\}$. We denote $I_j = (f(x_{j-1}),f(x_{j}))$ if $f(x_{j-1})<f(x_{j})$, or $I_j = (f(x_{j}),f(x_{j-1}))$ otherwise. Define $$ F(y) = \sum_{j=1}^n 1_{\{y\in I_j\}} = \big[\#\text{ of $j$ such that $y$ belongs to $I_j$}\big]. $$ Note that if $y\in I_j$, then by the Intermediate Value Theorem there exists $t_j \in (x_{j-1},x_j)$ such that $f(t_j)= y$. Since the number of such $t_j$ is at most $k$, this implies $F(y)\le k$ for each $y$. Because $I_j \subset [m,M]$ for each $j$, it follows that \begin{align*} \sum_{j=1}^n |f(x_j)-f(x_{j-1})| =& \sum_{j=1}^n \mathrm{len}(I_j)\\ =& \sum_{j=1}^n \int_{I_j}1\ \mathrm dy \\ =&\sum_{j=1}^n \int_m^M 1_{\{ y\in I_j\}} \mathrm dy\\ =& \int_m^M F(y) \mathrm dy \\ \le&\ k(M-m). \end{align*} And this implies $$ V_a^b(f) = \sup_{\Pi} \sum_{j=1}^n |f(x_j)-f(x_{j-1})| \le k(M-m).$$

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