Are continuous functions monotonic for very small ranges?

So I am wondering, if we have a continuous function $f : A \to B$, does a range $[x, x + h]$ exist for each $x\in A$ , $h = h(x)>0$ so that $f$ is monotonic in that range?

Solution 1:

No it is not true for example consider the function $x\sin(\frac{1}{x})$. It is continuous at zero if you take the limit to be the value of the function. But it oscillates very rapidly in every small neighbourhood around zero.

If I am not wrong even everywhere continuous but no where differentiable function has this property.I am referring to the function here

Solution 2:

And for an example that fails at every $c$, take the Weierstrass fuction defined by $$f(x)=\sum \limits_{n=0}^\infty\left(\dfrac 1 {2^n} \cos \left(15^n \pi x \right)\right).$$

The plot of this function exhibits self-similarity (see red circle below) and looks something like this:

Solution 3:

Monotone functions are differentiable almost everywhere. So if a function is monotonic on an interval it is differentiable on a set of positive measure. However there are continuous nowhere differentiable functions so it isn't true that continuous functions are monotonic in a (one sided) neighborhood of every point.

An example of a continuous nowhere differentiable function is the Takagi function.