# Examples where it's Riemann integrable but no derivative exists at pts

## Main Question or Discussion Point

What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?

HallsofIvy
Homework Helper
The function f(x)= 0 if $x\le 0$, 1 if x> 0 is integrable but has no derivative at x= 0. More generally, if f(x) has finite "jump" discontinuities at some points, it is still Riemann integrable but is not differentiable at those points.

or $$f(x)=\left| x\right|$$

HallsofIvy
Homework Helper
Those two examples also have the property that while $F(x)= \int f(t)dt$ is defined, F(x) itself has no dervative at x= 0.

mathwonk
Homework Helper
there are functions which are continuous everywhere hence integrable, but differentiable nowhere. perhaps the famous dirichlet function which equals zero at irrationals and 1/q at p/q is even differentiable nowhere. since it is continuous a.e. it is integrable.

$$V_{3}[\tex] arildno Science Advisor Homework Helper Gold Member Dearly Missed An example of a continuous (and hence integrable) function that is nowhere differentiable is the Weierstrass function: [tex]F(x)=\sum_{n=0}^{\infty}\frac{\sin((n!)^{2}x)}{n!}$$

Those two examples also have the property that while $F(x)= \int f(t)dt$ is defined, F(x) itself has no dervative at x= 0.
I'm not sure about this. Isn't $F(x)= \int_0^x |t|dt$ differentiable at 0? It is the piecewise function given by F(x)=x^2 for x>0 and F(x)=-x^2 for x>0 and F(0)=0.

HallsofIvy