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!
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!
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)...