1. ## Quick question

If a function is continuous on an interval and neither increasing or decreasing on any subinterval, is that the same as the function being nowhere differentiable on that interval? If not can you please elaborate on what the difference is

Thanks!

2. Originally Posted by HD09
If a function is continuous on an interval and neither increasing or decreasing on any subinterval
Isnt that function than constant on that interval... what means that function is differentiable on that interval

3. oops... I meant to specify besides the case where the function is just a constant

4. Originally Posted by HD09
oops... I meant to specify besides the case where the function is just a constant
give an example of that kind of function...
if function is not decreasing or increasing and continuous then it must be constant ( I think )

5. Originally Posted by josipive
give an example of that kind of function...
if function is not decreasing or increasing and continuous then it must be constant ( I think )
The Weierstrass function.

The derivative can never be non-zero, of course, because then for all sufficiently small $\displaystyle h_1,h_2>0$, $\displaystyle \frac{f(x+h_1)-f(x-h_2)}{h_1+h_2}$ will be always either positive (negative), so that on the interval $\displaystyle [h_1,h_2]$, $\displaystyle f$ is increasing (decreasing). This is just local linearity. Maybe we could have the derivative exist and be zero on a set of isolated points? $\displaystyle t \mapsto tW(t)$ is differentiable at $\displaystyle t=0$, ($\displaystyle W$ is the Weierstrass function)...