Let $f$ be a differentiable function on open interval $(a,b)$. Suppose $f'(x)$ is not identically zero. Show that there exists an subinterval $(c,d)$ such that $f(x)$ is strictly increasing or strictly decreasing on $(c,d)$.

How to prove this?

I think this statement is wrong...

  • 1
    $\begingroup$ $f'(x)$ is strictly... or $f(x)$ is strictly...? $\endgroup$ – mrs Mar 26 '13 at 7:19
  • $\begingroup$ I mean $f'(x)$ is strictly increasing... $\endgroup$ – XLDD Mar 26 '13 at 7:32
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    $\begingroup$ You must mean $f(x)$ strictly increasing or decreasing. Because on $(0,1)$ the function $f(x)=x$ has $f'(x)=1$ at all $x$, which is not strictly increasing or decreasing on any subinterval $(c,d)$ of $(0,1)$. $\endgroup$ – coffeemath Mar 26 '13 at 9:17
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    $\begingroup$ In relation to the comments under the answer by Taro, I would like to ask whether the question specifically wants to not assume that $f'$ is continuous. Whether or not that is assumed makes a large difference for the level of difficulty of the question. $\endgroup$ – Marc van Leeuwen Mar 26 '13 at 13:17
  • $\begingroup$ I'm trying to prove this without assuming the derivative is continuous, but first I gotta ask: is it actually true? $\endgroup$ – Javier Apr 8 '13 at 12:35

I show that it's true in case $f$ is not only differentiable but also its derivative is continuous.

Since $f'$ is non-zero function, there is a point $\tilde{x} \in (a, b)$ such that $f'(\tilde{x}) > 0$ or $f'(\tilde{x}) < 0$. Suppose $f'(\tilde{x}) > 0$. Then take $c$ and $d$ as follows: $$ \begin{align*} c &:= \inf\{\, \tilde{c} \mid a \leq \tilde{c} \leq \tilde{x}, \quad f'(c') > 0 \quad \text{for all $c' \in (\tilde{c}, \tilde{x}]$} \,\} \\ d &:= \sup\{\, \tilde{d} \mid \tilde{x} \leq \tilde{d} \leq b, \quad f'(d') > 0 \quad \text{for all $d' \in [\tilde{x}, \tilde{d})$} \,\}. \\ \end{align*}$$ From assumption that $f'$ is continuous, $c \neq \tilde{x} \neq d$. The interval $(c, d)$ is the required one (indeed, the largest interval containing $\tilde{x}$).

  • $\begingroup$ But this does not use that $f'$ is the derivative of something, the same would go for any continuous function in the place of $f'$. $\endgroup$ – Marc van Leeuwen Mar 26 '13 at 11:54
  • $\begingroup$ The required interval $(c, d)$ is the one which $f$ is strictly increasing for all $x \in (c, d)$ in the situation above. To show this, I showed $f'(x) > 0$ for all $x \in (c, d)$. This argument make sense since $f'$ is the derivative of $f$, doesn't it? $\endgroup$ – Orat Mar 26 '13 at 12:05
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    $\begingroup$ You are entirely correct. What I meant to say is that the question immediately translates into "if $f'$ not identically zero then either $f'(x)>0$ holds on some interval, or $f'(x)<0$ does"; since this statement is obviously true when replacing $f'$ by any continuous function, it is the case where $f'$ is not continuous that is the most interesting one. $\endgroup$ – Marc van Leeuwen Mar 26 '13 at 13:11
  • $\begingroup$ Wouldn't the fact that derivatives satisfy the intermediate value property suffice? $\endgroup$ – Javier Apr 7 '13 at 12:32
  • $\begingroup$ @JavierBadia I cannot see how to use it. Why not post it as an answer if the argument become much simpler? $\endgroup$ – Orat Apr 7 '13 at 21:00

The statement is indeed wrong. You can construct for example a function $f:\mathbb{R} \rightarrow\mathbb{R}$ which is differentiable everywhere such that both $\{x \in \mathbb{R} : f'(x) > 0\}$ and $\{x \in \mathbb{R} : f'(x) < 0\}$ are dense in $\mathbb R$ and thus $f$ is monotone on no interval. You can find such a construction on page 80 of A Second Course on Real Functions by van Rooij and Schikhof. See also here.


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