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I’m actually referring to another a question from here: Sufficient condition to show $f$ is monotonically increasing in some neighborhood

I know a counterexample has already been given, but I still do not fully understand why or how it is necessary for f’(x) to be continuous. Why does it have to be continuous in the whole interval, and not just at point x?

Actually I initially thought differentiability of f(x) was sufficient. Below is my reasoning.

Let $f : \left[a,b\right] \to \Bbb R$ be differentiable at some $c \in \left(a,b\right)$, with $f’(c) > 0$. Then, $$\lim_{x\to c}\frac{f(x) - f(c)}{x - c} = L \gt 0$$ So for $\epsilon = L \gt 0, \exists \delta \gt 0 :$
$$\left|x - c\right| \lt \delta \Rightarrow \left| \frac{f(x) - f(c)}{x - c} - L\right| \lt L \Rightarrow 0 \lt \frac{f(x) - f(c)}{x - c} \lt 2L$$ So when $-\delta \lt x - c \lt 0 \Rightarrow x \lt c$, and as $\frac{f(x) - f(c)}{x - c} \gt 0 \Rightarrow f(x) - f(c) \lt 0 \Rightarrow f(x) \lt f(c)$. Similarly when $0 \lt x - c \lt \delta \Rightarrow x \gt c$ and $f(x) \gt f(c)$. Hence we have found a neighbourhood $\left|x - c\right| \lt \delta$ in which $f(x)$ is increasing monotonically

Can someone please point out where I went wrong?

user63858
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    Something that jumps out to me immediately is that you haven't actually shown that $f$ is monotonic in the neighborhood. In particular, you need to show that for any $x,y \in (c- \delta, c + \delta)$, we have $x < y \Rightarrow f(x) < f(y)$. All you have done is shown that this works when comparing points in the interval to $c$. – Joe Jul 26 '18 at 18:04
  • @Joe oh right i just realised that. So the only way to show that the function is monotonically increasing is to prove that it has a positive derivative in that interval right? – user63858 Jul 26 '18 at 18:11
  • No not necessarily. Often times it's easier to show a function is monotone directly from the definition as opposed to finding the derivative. See the Cantor Function for example. – Joe Jul 26 '18 at 18:17

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